MGU Mathematics Series — Arithmetic and Pre-Algebra Foundations
Part III: Ratios, Percents, Exponents, and Scientific Notation
Powers of Ten · Conversion · Multiplication · Division · Real Data
Chapter 8 of 14
MGU Mathematics Series  ·  Arithmetic and Pre-Algebra Foundations  ·  Library Textbook Edition

Chapter 8:
Scientific
Notation

Powers of ten, conversion between standard and scientific form, multiplication and division, estimation, calculator use, and real-world applications in science and engineering

■ Math Level: Arithmetic and Pre-Algebra ■

■ 14 Sections  •  Part III of 5 ■

Follows Chapter 7: Exponents and Radicals  ·  Prepares students for Algebra I, Chemistry, and Physics

Chapter 8 — Front Matter

Purpose and Overview

Numbers in mathematics and everyday arithmetic are usually manageable. You might add grocery prices, measure a room, or calculate a tip, and the numbers stay within a comfortable range. But science, engineering, medicine, and technology regularly deal with numbers that are far outside that range, numbers so large or so small that writing them in ordinary form becomes impractical, confusing, and error-prone.

The distance from Earth to the nearest star outside our solar system is approximately 40,000,000,000,000 kilometers. A single human red blood cell is approximately 0.000008 meters in diameter. The number of atoms in a tablespoon of water is approximately 500,000,000,000,000,000,000,000. Writing these numbers in full is tedious at best and dangerously easy to misread at worst. A single missing zero changes the meaning entirely.

Scientific notation solves this problem. It is a compact, precise, internationally used system for writing very large and very small numbers using powers of ten. It does not change the value of a number. It changes the way the number is written so that it becomes easier to read, compare, compute with, and communicate.

This chapter teaches scientific notation from the ground up. You will begin by reviewing powers of ten, then learn to convert numbers into and out of scientific notation, then practice multiplying and dividing in scientific notation, and finally connect these skills to real scientific and technical data. By the end of this chapter, you will be able to read the language of scale that scientists, engineers, and mathematicians use every day.

Opening Question

A newspaper headline reads: "Scientists estimate the observable universe contains approximately 200,000,000,000 galaxies." A biology textbook states: "A typical bacterium is approximately 0.000001 meters long."

Both of these numbers are difficult to read at a glance. How many zeros are in the first number? How many decimal places is the second? Is there a better way to write them? There is. By the time you finish Chapter 8, you will be able to write both numbers in a form that is precise, compact, and easy to work with.

14Sections
12Objectives
10Key Terms
15+Worked Examples

Why This Chapter Matters

Scientific notation is not only a topic for science classes. It appears in astronomy, when describing distances between planets, stars, and galaxies; in chemistry, when describing the size of atoms and molecules; in biology, when describing bacteria, viruses, and cell dimensions; in medicine, when describing dosages and concentrations; in engineering, when working with tolerances and material properties; in computing, when describing file sizes and processor speeds; and in finance, when working with national debt figures and large-scale economic data.

Scientific notation also connects directly to the exponent rules you studied in Chapter 7, to decimal place value from Chapter 5, and to the estimation skills you have been building throughout this textbook. It is a natural continuation of the number sense this course has been developing from the beginning.

Learning Objectives

  1. 1Explain why scientific notation is a useful tool for writing very large and very small numbers.
  2. 2Compute positive and negative integer powers of ten.
  3. 3Identify the form of a number written in correct scientific notation.
  4. 4Convert standard decimal numbers into scientific notation.
  5. 5Convert numbers written in scientific notation back into standard decimal form.
  6. 6Multiply two numbers written in scientific notation.
  7. 7Divide two numbers written in scientific notation.
  8. 8Perform combined multiplication and division with scientific notation.
  9. 9Estimate using scientific notation and order-of-magnitude reasoning.
  10. 10Interpret scientific notation as it appears on a calculator.
  11. 11Read and compare real scientific data written in scientific notation.
  12. 12Describe practical applications of scientific notation in science and engineering.
Chapter 8 — Contents

Sections at a Glance

Sections 8.1 – 8.4
Powers of Ten: The Foundation of Scale
Why We Need Scientific Notation · Powers of Ten · Positive Exponents · Negative Exponents
Section 8.1

Why Scientific Notation Is Needed

The number system you use every day is built on base ten. Place value in its ordinary written form, however, becomes clumsy when the numbers grow very large or very small. Consider the following examples from actual science.

QuantityStandard Form
Distance to Proxima Centauri (km)40,000,000,000,000
Mass of Earth (kg)5,970,000,000,000,000,000,000,000
Diameter of a hydrogen atom (m)0.0000000001
Mass of a proton (kg)0.000000000000000000000000001673

Each of these numbers is technically writable in standard form. But try reading the mass of a proton aloud. Try copying it from one line to another without losing track of the zeros. Try comparing it to a similar number to see which is larger. The difficulty is not with the mathematics. The difficulty is with the notation itself.

Scientific notation addresses this by using the structure of powers of ten to compress the writing of any number into a short, readable, precise form. A number written in scientific notation always takes this form:

\[ a \times 10^n \]

where \(a\) is a decimal number satisfying \(1 \leq a < 10\), and \(n\) is an integer (positive, negative, or zero). The coefficient \(a\) captures the significant digits. The power of ten \(10^n\) captures the scale.

QuantityScientific Notation
Distance to Proxima Centauri (km)\(4 \times 10^{13}\)
Mass of Earth (kg)\(5.97 \times 10^{24}\)
Diameter of a hydrogen atom (m)\(1 \times 10^{-10}\)
Mass of a proton (kg)\(1.673 \times 10^{-27}\)
Student Note — The Form Is Strict

The form \(a \times 10^n\) is not optional. A number like \(23 \times 10^4\) is not in proper scientific notation, even though it uses a power of ten. The coefficient \(a\) must be at least 1 and less than 10. You must adjust the coefficient until it is in the correct range.

Practice 8.1Concept Check
  1. In your own words, explain why scientific notation is more useful than standard form for numbers like the mass of a proton or the distance to a star.
  2. Which of the following are in proper scientific notation? For each one that is not, explain why.

    a. \(4.5 \times 10^6\)     b. \(30 \times 10^3\)     c. \(0.7 \times 10^{-2}\)     d. \(1.0 \times 10^0\)     e. \(9.99 \times 10^5\)
  3. A classmate says, "A negative exponent means the number is negative." Write a counterexample that shows this statement is false.
Section 8.2

Powers of Ten

A power of ten is ten raised to an integer exponent. You studied exponents in Chapter 7. Here you apply that knowledge specifically to base ten, which is especially clean because our entire number system is built around it.

\[ 10^1 = 10 \qquad 10^2 = 100 \qquad 10^3 = 1{,}000 \qquad 10^4 = 10{,}000 \]
\[ 10^{-1} = 0.1 \qquad 10^{-2} = 0.01 \qquad 10^{-3} = 0.001 \]
\[ 10^0 = 1 \]

Each increase by one in the exponent multiplies the value by ten, producing a new zero in standard form. Each decrease by one in the exponent divides the value by ten, moving the significant digit one more place to the right of the decimal point. The zero exponent fits the pattern exactly: moving from \(10^1 = 10\) down to \(10^0 = 1\) is a single division by ten.

PowerStandard FormName
\(10^{12}\)1,000,000,000,000one trillion
\(10^9\)1,000,000,000one billion
\(10^6\)1,000,000one million
\(10^3\)1,000one thousand
\(10^0\)1one
\(10^{-3}\)0.001one thousandth
\(10^{-6}\)0.000001one millionth
\(10^{-9}\)0.000000001one billionth
MGU Library Connection
  • The formula sheet in Appendix E includes a full table of powers of ten from \(10^{-12}\) to \(10^{12}\). Refer to it as needed.
  • Chapter 7: Exponents and Radicals — reviews the product, power, and quotient rules used in Sections 8.8–8.10.
Practice 8.2Powers of Ten — Skill Practice

Section A — Evaluate each power of ten in standard form.

  1. \(10^4\)
  2. \(10^8\)
  3. \(10^{-2}\)
  4. \(10^{-6}\)
  5. \(10^0\)
  6. \(10^{11}\)
  7. \(10^{-9}\)
  8. \(10^7\)

Section B — Write each standard form value as a power of ten.

  1. 1,000
  2. 100,000,000
  3. 0.01
  4. 0.0000001
  5. 1,000,000,000,000
  6. 0.000000001
  7. 10
  8. 1
Section 8.3

Positive Powers of Ten

Positive powers of ten represent numbers greater than or equal to one. The larger the exponent, the larger the number. When a number is written as \(a \times 10^n\) with a positive \(n\), the number is at least ten.

Multiplying any number by \(10^n\) (where \(n\) is a positive integer) moves the decimal point \(n\) places to the right. The exponent tells you the order of magnitude, the approximate power of ten that describes the scale of the number.

Example 8.3.1
Evaluating a positive power of ten
  • ProblemEvaluate \(10^7\) in standard form.
  • ThinkIn standard form, a power of ten is 1 followed by that many zeros.
  • Solve10^7 = 10,000,000
  • CheckCount the zeros. There are seven zeros after the 1. This matches the exponent.
Answer: \(10^7 = 10{,}000{,}000\)
Example 8.3.2
Multiplying by a positive power of ten
  • ProblemCompute \(2.8 \times 10^4\).
  • ThinkMultiplying by \(10^4\) moves the decimal point four places to the right.
  • Solve2.8 × 10^4 → 28,000
  • Check\(2.8 \times 10{,}000 = 28{,}000\). Correct.
Answer: \(28{,}000\)
Practice 8.3Positive Powers — Compute each product.
  1. \(4.0 \times 10^3\)
  2. \(2.7 \times 10^5\)
  3. \(6.0 \times 10^7\)
  4. \(1.45 \times 10^5\)
  5. \(3.0 \times 10^9\)
  6. \(9.1 \times 10^{11}\)
Section 8.4

Negative Powers of Ten

Negative powers of ten represent numbers between zero and one. The more negative the exponent, the smaller the number. Multiplying any number by \(10^{-n}\) (where \(n\) is a positive integer) moves the decimal point \(n\) places to the left.

Critical Distinction

A negative exponent does not mean the number is negative. It means the number is small, specifically a fraction of one. The number \(6.02 \times 10^{-3} = 0.00602\) is a small positive number. The negative sign belongs to the exponent on the ten, not to the value itself.

Example 8.4.1
Evaluating a negative power of ten
  • ProblemEvaluate \(10^{-5}\) in standard form.
  • ThinkA negative power of ten means \(\frac{1}{10^5}\). The significant digit 1 sits five places to the right of the decimal point.
  • Solve10^{-5} = 1/100,000 = 0.00001
  • CheckCount: the 1 is in the fifth decimal place. Correct.
Answer: \(10^{-5} = 0.00001\)
Example 8.4.2
Multiplying by a negative power of ten
  • ProblemCompute \(4.5 \times 10^{-4}\).
  • ThinkMultiplying by \(10^{-4}\) moves the decimal point four places to the left.
  • Solve4.5 × 10^{-4} → 0.00045
  • Check\(4.5 \div 10{,}000 = 0.00045\). Correct.
Answer: \(0.00045\)
Practice 8.4Negative Powers — Compute each product.
  1. \(1.5 \times 10^{-2}\)
  2. \(8.3 \times 10^{-4}\)
  3. \(3.14 \times 10^{-6}\)
  4. \(6.02 \times 10^{-3}\)
  5. \(9.0 \times 10^{-8}\)
  6. \(2.5 \times 10^{-10}\)
Sections 8.5 – 8.7
Standard Form and Scientific Notation: Converting Both Ways
What Makes Notation Correct · Standard to Scientific · Scientific to Standard
Section 8.5

Standard Form and Scientific Notation

Every number can be written in two forms. Standard form is the familiar decimal number with all digits in place, such as 93,000,000 or 0.000052. Scientific notation is a compact form using a coefficient between 1 and 10 multiplied by a power of ten, such as \(9.3 \times 10^7\) or \(5.2 \times 10^{-5}\). Both forms represent the same value. Scientific notation neither changes the number nor rounds it, unless rounding is explicitly done.

Three Conditions for Proper Scientific Notation

1. The coefficient is a decimal number satisfying \(1 \leq a < 10\).

2. The coefficient is multiplied by a power of ten.

3. The exponent is an integer (positive, negative, or zero).

ExpressionProper?Reason if Not
\(3.0 \times 10^8\)Yes—
\(7.41 \times 10^5\)Yes—
\(23 \times 10^4\)NoCoefficient 23 is not less than 10
\(0.6 \times 10^3\)NoCoefficient 0.6 is not at least 1
\(5 \times 3^4\)NoBase is not 10
Order of Magnitude

The power of ten in scientific notation is called the order of magnitude. It gives you an immediate sense of the size of the number. A number with \(10^6\) is in the millions. A number with \(10^{-9}\) is in the billionths. Reading the exponent at a glance is one of the main practical advantages of scientific notation.

Practice 8.5Identifying Proper Scientific Notation

For each expression, state whether it is in proper scientific notation. If not, identify the problem.

  1. \(4.5 \times 10^6\)
  2. \(30 \times 10^3\)
  3. \(0.7 \times 10^{-2}\)
  4. \(1.0 \times 10^0\)
  5. \(9.99 \times 10^{-4}\)
  6. \(10 \times 10^8\)
  7. \(1.602 \times 10^{-19}\)
  8. \(0.001 \times 10^5\)
Section 8.6

Converting Standard Numbers to Scientific Notation

Five-Step Conversion Procedure

Step 1. Locate the decimal point. (If none is written, it sits at the right end of the number.)

Step 2. Move the decimal point left or right until you have a number between 1 and 10.

Step 3. Count how many places you moved. This count becomes the exponent.

Step 4. Determine the sign: moved left → positive exponent; moved right → negative exponent.

Step 5. Write in the form \(a \times 10^n\).

Original NumberDecimal MovedExponent Sign
Greater than or equal to 10LeftPositive
Between 0 and 1RightNegative
Example 8.6.1
Large number to scientific notation
  • ProblemWrite 47,300,000 in scientific notation.
  • ThinkThe decimal point is at the right end. Move it left until the coefficient is between 1 and 10.
  • Solve47,300,000. → 4.73 Decimal moved 7 places left → exponent is +7
  • Check\(4.73 \times 10^7 = 4.73 \times 10{,}000{,}000 = 47{,}300{,}000\). Correct.
Answer: \(4.73 \times 10^7\)
Example 8.6.2
Small number to scientific notation
  • ProblemWrite 0.000059 in scientific notation.
  • ThinkThe number is between 0 and 1. Move the decimal point right until the coefficient is between 1 and 10.
  • Solve0.000059 → 5.9 Decimal moved 5 places right → exponent is -5
  • Check\(5.9 \times 10^{-5} = 5.9 \times 0.00001 = 0.000059\). Correct.
Answer: \(5.9 \times 10^{-5}\)
Example 8.6.3
Correcting improper scientific notation
  • ProblemThe number \(52 \times 10^3\) is not in proper form. Rewrite it correctly.
  • ThinkThe coefficient 52 is greater than 10. Move its decimal one place left to get 5.2. To keep the value the same, increase the exponent by 1.
  • Solve52 × 10^3 = 5.2 × 10^1 × 10^3 = 5.2 × 10^4
  • Check\(5.2 \times 10^4 = 52{,}000\) and \(52 \times 10^3 = 52{,}000\). Same value. Correct.
Answer: \(5.2 \times 10^4\)
Adjusting Coefficients — Keep the Value Constant

When you adjust an improper coefficient, always adjust the exponent in the opposite direction to keep the value the same. Moving the decimal left (making the coefficient smaller) increases the exponent. Moving the decimal right (making the coefficient larger) decreases the exponent.

Practice 8.6Write each standard form number in scientific notation.
  1. 5,000
  2. 780,000
  3. 43,200,000
  4. 9,100,000,000
  5. 0.006
  6. 0.000041
  7. 0.00000307
  8. 602,000,000,000,000,000,000,000
  9. 0.000000000000167
  10. 50,000,000,000

Correct each expression — rewrite in proper scientific notation.

  1. \(32 \times 10^5\)
  2. \(0.8 \times 10^4\)
  3. \(14.7 \times 10^{-2}\)
  4. \(100 \times 10^3\)
  5. \(0.025 \times 10^9\)
Section 8.7

Converting Scientific Notation to Standard Form

Converting from scientific notation back to standard form means expanding the number by following the instruction of the exponent. A positive exponent tells you to move the decimal point that many places to the right, making the number larger. A negative exponent tells you to move the decimal point that many places to the left, making the number smaller. Fill in zeros as needed when you run out of digits.

Example 8.7.1
Positive exponent to standard form
  • ProblemWrite \(3.15 \times 10^6\) in standard form.
  • ThinkExponent is 6: move the decimal point in 3.15 six places to the right.
  • Solve3.15 → 31.5 → 315 → 3150 → 31500 → 315000 → 3,150,000
  • Check3,150,000 is in the millions, which matches \(10^6\). Correct.
Answer: \(3{,}150{,}000\)
Example 8.7.2
Negative exponent to standard form
  • ProblemWrite \(8.4 \times 10^{-7}\) in standard form.
  • ThinkExponent is \(-7\): move the decimal point in 8.4 seven places to the left.
  • Solve8.4 → 0.84 → 0.084 → 0.0084 → 0.00084 → 0.000084 → 0.0000084 → 0.00000084
  • CheckThe first significant digit (8) sits seven places to the right of the decimal point. Correct.
Answer: \(0.00000084\)
Common Directional Error

Students sometimes confuse the direction of decimal movement. Remember: positive exponent → move right → number gets bigger. Negative exponent → move left → number gets smaller. If you end up with a tiny number after applying a positive exponent, recheck the direction.

Practice 8.7Write each expression in standard form.
  1. \(3 \times 10^4\)
  2. \(7.2 \times 10^6\)
  3. \(1.85 \times 10^9\)
  4. \(4.0 \times 10^{-3}\)
  5. \(6.02 \times 10^{-7}\)
  6. \(9.5 \times 10^{10}\)
  7. \(2.22 \times 10^{-5}\)
  8. \(1.0 \times 10^{-12}\)
Sections 8.8 – 8.11
Computing and Estimating in Scientific Notation
Multiplication · Division · Combined Operations · Estimation
Section 8.8

Multiplying Scientific Notation

Because scientific notation is built on powers of ten, and because you know the product rule for exponents from Chapter 7, multiplication in scientific notation is clean and structured. Recall:

\[ 10^m \times 10^n = 10^{m+n} \]
Multiplication Procedure

Step 1. Multiply the two coefficients together.

Step 2. Add the exponents of the powers of ten.

Step 3. Adjust if the new coefficient is not between 1 and 10.

(a × 10ᵐ) × (b × 10ⁿ) = (a × b) × 10ᵐ⁺ⁿ

Example 8.8.1
Basic multiplication in scientific notation
  • ProblemCompute \((3 \times 10^4) \times (2 \times 10^5)\).
  • ThinkMultiply the coefficients; add the exponents.
  • Solve(3 × 2) × 10^(4+5) = 6 × 10^9
  • Check\(30{,}000 \times 200{,}000 = 6{,}000{,}000{,}000 = 6 \times 10^9\). Correct.
Answer: \(6 \times 10^9\)
Example 8.8.2
Adjusting after multiplication
  • ProblemCompute \((4 \times 10^3) \times (5 \times 10^6)\).
  • ThinkPreliminary: \((4 \times 5) \times 10^{3+6} = 20 \times 10^9\). Coefficient 20 is too large. Adjust.
  • Solve20 × 10^9 = 2 × 10^1 × 10^9 = 2 × 10^10
  • Check\(4{,}000 \times 5{,}000{,}000 = 20{,}000{,}000{,}000 = 2 \times 10^{10}\). Correct.
Answer: \(2 \times 10^{10}\)
Example 8.8.3
Multiplication with negative exponents
  • ProblemCompute \((6 \times 10^{-3}) \times (3 \times 10^{-4})\).
  • ThinkCoefficients: \(6 \times 3 = 18\). Exponents: \(-3 + (-4) = -7\). Adjust 18.
  • Solve18 × 10^{-7} = 1.8 × 10^1 × 10^{-7} = 1.8 × 10^{-6}
  • Check\(0.006 \times 0.0001 = 0.0000006 = 6 \times 10^{-7}\). With more precise coefficients: \(1.8 \times 10^{-6} = 0.0000018\). Correct.
Answer: \(1.8 \times 10^{-6}\)
Practice 8.8Multiply. Write each answer in proper scientific notation.
  1. \((2 \times 10^3) \times (3 \times 10^4)\)
  2. \((5 \times 10^6) \times (4 \times 10^2)\)
  3. \((7 \times 10^{-2}) \times (2 \times 10^5)\)
  4. \((3 \times 10^{-4}) \times (6 \times 10^{-3})\)
  5. \((1.5 \times 10^4) \times (2 \times 10^3)\)
  6. \((4.0 \times 10^7) \times (2.5 \times 10^{-2})\)
  7. \((8 \times 10^5) \times (5 \times 10^4)\)
  8. \((3.2 \times 10^{-6}) \times (4.0 \times 10^8)\)
Section 8.9

Dividing Scientific Notation

Division in scientific notation follows the same logic as multiplication but uses the quotient rule for exponents from Chapter 7:

\[ \frac{10^m}{10^n} = 10^{m-n} \]
Division Procedure

Step 1. Divide the two coefficients.

Step 2. Subtract the exponent of the denominator from the exponent of the numerator.

Step 3. Adjust if the coefficient is not between 1 and 10.

(a × 10ᵐ) ÷ (b × 10ⁿ) = (a ÷ b) × 10ᵐ⁻ⁿ

Example 8.9.1
Basic division in scientific notation
  • ProblemCompute \(\dfrac{8 \times 10^9}{2 \times 10^3}\).
  • ThinkDivide coefficients: \(8 \div 2 = 4\). Subtract exponents: \(9 - 3 = 6\).
  • Solve4 × 10^6
  • Check\(8{,}000{,}000{,}000 \div 2{,}000 = 4{,}000{,}000 = 4 \times 10^6\). Correct.
Answer: \(4 \times 10^6\)
Example 8.9.2
Division yielding a negative exponent
  • ProblemCompute \(\dfrac{9 \times 10^2}{3 \times 10^7}\).
  • ThinkDivide coefficients: \(9 \div 3 = 3\). Subtract exponents: \(2 - 7 = -5\).
  • Solve3 × 10^{-5}
  • Check\(900 \div 30{,}000{,}000 = 0.00003 = 3 \times 10^{-5}\). Correct.
Answer: \(3 \times 10^{-5}\)
Example 8.9.3
Division requiring coefficient adjustment
  • ProblemCompute \(\dfrac{5 \times 10^8}{4 \times 10^3}\).
  • ThinkDivide coefficients: \(5 \div 4 = 1.25\). Subtract exponents: \(8 - 3 = 5\). Coefficient 1.25 is already in range. No adjustment needed.
  • Solve1.25 × 10^5
  • Check\(500{,}000{,}000 \div 4{,}000 = 125{,}000 = 1.25 \times 10^5\). Correct.
Answer: \(1.25 \times 10^5\)
Subtract the Exponents — Do Not Divide Them

When dividing in scientific notation, the exponents are subtracted, not divided. Students occasionally try to divide the exponents just as they divide the coefficients. Keep the operations separate: divide the coefficients and subtract the exponents.

Practice 8.9Divide. Write each answer in proper scientific notation.
  1. \(\dfrac{6 \times 10^8}{2 \times 10^3}\)
  2. \(\dfrac{9 \times 10^7}{3 \times 10^4}\)
  3. \(\dfrac{4 \times 10^3}{8 \times 10^6}\)
  4. \(\dfrac{5 \times 10^{-2}}{2 \times 10^3}\)
  5. \(\dfrac{7.2 \times 10^9}{1.2 \times 10^5}\)
  6. \(\dfrac{3.0 \times 10^{-4}}{6.0 \times 10^{-9}}\)
Section 8.10

Multiplying and Dividing Scientific Notation Together

Some calculations require both multiplication and division in the same expression. This is common in physics formulas, chemistry calculations, and engineering unit conversions. The approach is the same: handle the coefficients with arithmetic and handle the powers of ten with exponent rules, then combine and adjust.

Example 8.10.1
Combined multiplication and division
  • ProblemCompute \(\dfrac{(6 \times 10^5) \times (4 \times 10^3)}{8 \times 10^4}\).
  • ThinkMultiply the numerator first, then divide by the denominator.
  • Solve Step 1 — Numerator: (6 × 4) × 10^(5+3) = 24 × 10^8 Step 2 — Divide: (24 ÷ 8) × 10^(8-4) = 3 × 10^4
  • Check\(600{,}000 \times 4{,}000 = 2{,}400{,}000{,}000\). Divided by \(80{,}000\): \(= 30{,}000 = 3 \times 10^4\). Correct.
Answer: \(3 \times 10^4\)
Example 8.10.2
Combined operations with negative exponents
  • ProblemCompute \(\dfrac{(2 \times 10^{-3}) \times (6 \times 10^5)}{4 \times 10^{-2}}\).
  • Solve Step 1 — Numerator: (2 × 6) × 10^(-3+5) = 12 × 10^2 = 1.2 × 10^3 Step 2 — Divide: (1.2 ÷ 4) × 10^(3-(-2)) = 0.3 × 10^5 Step 3 — Adjust: 0.3 × 10^5 = 3 × 10^{-1} × 10^5 = 3 × 10^4
Answer: \(3 \times 10^4\)
Subtracting a Negative Exponent

Remember that subtracting a negative exponent changes the sign: \(3 - (-2) = 3 + 2 = 5\). This step causes many errors in multi-part problems. Write it out carefully.

Practice 8.10Combined operations. Write each answer in proper scientific notation.
  1. \(\dfrac{(3 \times 10^4) \times (4 \times 10^5)}{6 \times 10^3}\)
  2. \(\dfrac{(8 \times 10^{-3}) \times (2 \times 10^7)}{4 \times 10^2}\)
  3. \(\dfrac{(5 \times 10^6) \times (6 \times 10^{-4})}{1.5 \times 10^4}\)
  4. \(\dfrac{(9 \times 10^{11}) \times (2 \times 10^{-5})}{3 \times 10^8}\)
Section 8.11

Estimating with Scientific Notation

In scientific and engineering work, an exact answer is not always necessary or even possible. Scientific notation is excellent for quick estimation because the exponent immediately tells you the order of magnitude. The simplest form of estimation is to round the coefficients and add or subtract the exponents.

Comparing two numbers in scientific notation is also straightforward: compare the exponents first. If the exponents are different, the number with the larger exponent is larger, regardless of the coefficients. If the exponents are equal, compare the coefficients.

Example 8.11.1
Estimating a product
  • ProblemEstimate \((2.9 \times 10^8) \times (6.1 \times 10^5)\) to one significant figure.
  • ThinkRound 2.9 to 3 and round 6.1 to 6. Multiply rounded coefficients and add exponents.
  • Solve(3 × 10^8) × (6 × 10^5) = 18 × 10^13 ≈ 2 × 10^14
Answer: approximately \(2 \times 10^{14}\)   (exact: \(1.769 \times 10^{14}\))
Example 8.11.2
Comparing two measurements
  • ProblemA bacterium is \(2.3 \times 10^{-6}\) meters long. A virus is \(8.0 \times 10^{-9}\) meters long. Approximately how many times larger is the bacterium?
  • Solve 2.3 × 10^{-6} / 8.0 × 10^{-9} ≈ (2 / 8) × 10^{-6-(-9)} = 0.25 × 10^3 = 2.5 × 10^2
Answer: The bacterium is approximately 250 times larger than the virus.
Practice 8.11Estimation Problems

Use rounded coefficients and order-of-magnitude reasoning. You do not need exact answers.

  1. Estimate \((3.1 \times 10^5) \times (2.9 \times 10^4)\) to one significant figure.
  2. Estimate \((7.8 \times 10^9) \div (4.1 \times 10^3)\) to one significant figure.
  3. Estimate \((5.6 \times 10^{-4}) \times (3.2 \times 10^{-5})\) to one significant figure.
  4. Two quantities are \(6.2 \times 10^{14}\) and \(4.8 \times 10^{11}\). Without computing, identify which is larger and estimate approximately how many times larger it is.
  5. The human brain contains approximately \(8.6 \times 10^{10}\) neurons. Each neuron has approximately \(7 \times 10^3\) synaptic connections. Estimate the total number of synaptic connections in the brain.
Sections 8.12 – 8.14
Calculators, Real Data, and the Road Ahead
Calculator Notation · Real Scientific Data · Preparing for Science and Algebra
Section 8.12

Scientific Notation on Calculators

Most calculators cannot display the full multiplication sign and raised exponent of scientific notation on a single screen. Instead, they use the letter E (sometimes EE, exp, or e) to stand for "times ten to the power of." On a calculator, the display 3.7E8 means \(3.7 \times 10^8\), and 5.2E-4 means \(5.2 \times 10^{-4}\).

Calculator DisplayScientific NotationStandard Form
4.5E7\(4.5 \times 10^7\)45,000,000
1.2E-5\(1.2 \times 10^{-5}\)0.000012
6.674E-11\(6.674 \times 10^{-11}\)0.00000000006674
9.461E15\(9.461 \times 10^{15}\)9,461,000,000,000,000

Entering Scientific Notation on a Calculator

The general method: (1) type the coefficient; (2) press the EE, E, or EXP key, not the multiplication key and not the letter E for variables; (3) type the exponent. For a negative exponent, press the negative key, not the subtraction key, before typing the exponent digits.

Negative Key vs. Subtraction Key

On many calculators, there is a difference between the subtraction key and the negative (change sign) key. The subtraction key is a binary operation between two numbers. The negative key makes a number negative. For entering negative exponents in scientific notation, you need the negative key, not the subtraction key. Using the wrong key is a common input error that produces a completely different calculation.

MGU Library Connection
  • Appendix H: Calculator Skills — covers scientific calculator use in more detail, including exponent, fraction, and order of operations problems.
Practice 8.12Calculator Notation

Convert each calculator display to proper scientific notation, then to standard form.

  1. 2.5E4
  2. 8.0E-6
  3. 1.602E-19
  4. 6.022E23
  5. 3.0E8
  6. 9.8E-3
Section 8.13

Scientific Notation in Real Data

Scientific notation appears regularly in science textbooks, lab reports, news articles, and data tables. The ability to read, interpret, and compare numbers in scientific notation is a practical skill, not just an academic one. Here are values from fields where students are likely to encounter scientific notation first.

FieldQuantityValue
AstronomySpeed of light in vacuum\(3.0 \times 10^8\) m/s
AstronomyEarth–Sun distance\(1.5 \times 10^{11}\) m
AstronomyAge of the universe\(1.38 \times 10^{10}\) years
ChemistryAvogadro's number\(6.022 \times 10^{23}\)
PhysicsCharge of one electron\(1.6 \times 10^{-19}\) coulombs
PhysicsMass of a proton\(1.673 \times 10^{-27}\) kg
BiologyDiameter of a human cell\(1 \times 10^{-5}\) m
BiologyNeurons in human brain\(8.6 \times 10^{10}\)
ComputingOne terabyte (bytes)\(1 \times 10^{12}\)
FinanceU.S. GDP (approximate)\(2.5 \times 10^{13}\) dollars
Example 8.13.1
Comparing two physical constants
  • ProblemA proton has mass \(1.673 \times 10^{-27}\) kg. An electron has mass \(9.109 \times 10^{-31}\) kg. Approximately how many times more massive is a proton?
  • Solve 1.673 × 10^{-27} / 9.109 × 10^{-31} ≈ (1.7 / 9) × 10^{-27-(-31)} = 0.189 × 10^4 ≈ 1.9 × 10^3
Answer: A proton is approximately 1,836 times more massive than an electron (about \(1.9 \times 10^3\) by our rounded estimate).
Example 8.13.2
Expressing a real measurement in scientific notation
  • ProblemThe distance from Earth to Proxima Centauri is approximately 40,208,000,000,000 kilometers. Write this in scientific notation.
  • Solve40,208,000,000,000 ≈ 4.0 × 10^13 km (to two significant figures)
Answer: approximately \(4.0 \times 10^{13}\) kilometers.
Practice 8.13Science Applications

Use the data table above or the values provided. Write all answers in scientific notation.

  1. A spacecraft travels at \(2.0 \times 10^4\) meters per second. How many seconds would it take to travel the Earth–Sun distance? Write your answer in scientific notation.
  2. How many times greater is the speed of light than the spacecraft in Problem 1?
  3. The mass of one hydrogen atom is approximately \(1.67 \times 10^{-27}\) kg. Avogadro's number is \(6.022 \times 10^{23}\). What is the approximate total mass of one mole of hydrogen atoms?
  4. A red blood cell is approximately \(8 \times 10^{-6}\) meters wide. A strand of DNA is approximately \(2 \times 10^{-9}\) meters wide. How many times wider is the red blood cell?
  5. A new storage system holds \(3.5 \times 10^3\) terabytes. One terabyte is \(10^{12}\) bytes. How many bytes does the system hold?
Section 8.14

Preparing for Science and Engineering Applications

Scientific notation is a foundational tool for every quantitative field of study. In this chapter you have learned why very large and very small numbers need a compact notation, how powers of ten encode size and scale, how to convert numbers in both directions, how to multiply and divide using exponent rules, how to estimate using orders of magnitude, how calculators display and accept scientific notation, and how to read and compare real scientific data.

These skills connect directly to several future areas of study. In Algebra I and Algebra II, you will work with scientific notation inside equations, use it to interpret function values, and apply it in formulas. The exponent rules from Chapter 7 and Chapter 8 will reappear regularly in polynomial expressions and exponential functions.

In MGU science courses, scientific notation will be used from the first page. Chemistry calculations involving Avogadro's number, physics problems involving gravitational constants and electrical charges, biology measurements of cells and organelles, and astronomy calculations involving distances and timescales all rely on your ability to read, write, and compute in scientific notation fluently.

One of the most useful things scientific notation gives you is a sense of scale, an intuitive feel for how large or small the numbers you encounter actually are. When you know that a human hair is roughly \(7 \times 10^{-5}\) meters wide and a virus is roughly \(1 \times 10^{-8}\) meters wide, you can immediately see that the hair is several thousand times wider than the virus. You do not need to compute the exact ratio to have a meaningful understanding of the difference. Mathematics at every level is partly about building this kind of number sense, and scientific notation is one of the most direct tools we have for developing it.

MGU Library Connection
  • Appendix E: Formula Sheet — includes standard scientific constants in scientific notation.
  • Appendix J: Algebra I Readiness Check — includes problems using scientific notation in equation and formula contexts.
  • MGU Algebra I, Chapter 2: Real Numbers and Algebraic Properties — builds on the number sense developed here.
  • MGU Dictionary entries: scientific notation, order of magnitude, exponent, power of ten, coefficient.
Chapter 8 — Back Matter

Chapter Summary

Key Terms Review

scientific notation
The form \(a \times 10^n\) where \(1 \leq a < 10\) and \(n\) is an integer. A compact, precise way to write very large or very small numbers.
standard form
An ordinary decimal number written without exponents, such as 4,500,000 or 0.000032.
coefficient
In scientific notation, the number between 1 (inclusive) and 10 (exclusive) that is multiplied by the power of ten.
power of ten
A number expressed as \(10^n\) for some integer \(n\). The building block of scientific notation.
positive exponent
An exponent greater than zero. In scientific notation, indicates a number greater than or equal to the coefficient.
negative exponent
An exponent less than zero. In scientific notation, indicates a number smaller than the coefficient. Does not make the number negative.
order of magnitude
The power of ten that describes the scale of a number. The exponent in scientific notation. Used to compare sizes quickly.
decimal point movement
The process of moving the decimal point when converting between standard form and scientific notation, or when multiplying or dividing by powers of ten.
E notation
The way calculators display scientific notation, using the letter E instead of \(\times 10\). For example, 3.7E8 means \(3.7 \times 10^8\).
estimation
An approximate calculation using rounded values. In scientific notation, often means rounding coefficients and reading the order of magnitude directly from the exponent.

Concept Review Questions

Answer the following questions in complete sentences.

  1. Explain in your own words why scientific notation is more useful than standard form for writing numbers like the mass of a proton or the distance to a star.
  2. What two conditions must the coefficient of a number in scientific notation satisfy?
  3. How do you know whether the exponent will be positive or negative when converting a standard form number to scientific notation?
  4. A classmate writes \(15 \times 10^6\) and says it is in scientific notation. Explain the error and correct the expression.
  5. How does the order of magnitude help you compare two numbers written in scientific notation without fully computing them?
  6. Explain the difference between the way scientific notation is written by hand and the way a calculator displays it.
  7. Why does multiplying by \(10^{-4}\) make a number smaller, even though you are multiplying?
  8. How are the exponent rules from Chapter 7 used in multiplying and dividing scientific notation?
  9. Give one example from science and one from everyday life where scientific notation would be more useful than standard form.
  10. Describe in your own words what it means for a number to have an order of magnitude of \(10^9\).

Skill Practice

Part APowers of Ten

Evaluate each power of ten in standard form.

  1. \(10^5\)
  2. \(10^{-3}\)
  3. \(10^0\)
  4. \(10^{10}\)
  5. \(10^{-7}\)
  6. \(10^{12}\)
  7. \(10^{-9}\)
  8. \(10^2\)
Part BConverting to Scientific Notation
  1. 860,000
  2. 0.0000034
  3. 72,000,000,000
  4. 0.000000009
  5. 450
  6. 0.000000000000167
  7. 1,380,000,000,000
  8. 0.00000000009461
Part CConverting to Standard Form
  1. \(5 \times 10^6\)
  2. \(3.1 \times 10^{-4}\)
  3. \(7.02 \times 10^8\)
  4. \(9.0 \times 10^{-11}\)
  5. \(1.5 \times 10^3\)
  6. \(4.44 \times 10^{-7}\)
  7. \(8.0 \times 10^{13}\)
  8. \(2.718 \times 10^{-2}\)
Part DMultiplication
  1. \((4 \times 10^3) \times (3 \times 10^5)\)
  2. \((2 \times 10^{-2}) \times (5 \times 10^6)\)
  3. \((6 \times 10^4) \times (7 \times 10^3)\)
  4. \((1.5 \times 10^{-5}) \times (4.0 \times 10^2)\)
Part EDivision
  1. \(\dfrac{9 \times 10^8}{3 \times 10^5}\)
  2. \(\dfrac{6 \times 10^{-3}}{2 \times 10^4}\)
  3. \(\dfrac{8 \times 10^7}{4 \times 10^{-3}}\)
  4. \(\dfrac{5 \times 10^{-6}}{2.5 \times 10^{-9}}\)

Word Problems

Write all answers in scientific notation. Show the setup and working for each problem.

  1. A strand of spider silk is approximately \(3 \times 10^{-6}\) meters in diameter. A human hair is approximately \(7 \times 10^{-5}\) meters in diameter. How many times wider is a human hair than a strand of spider silk?
  2. A computer processor performs \(4 \times 10^9\) operations per second. How many operations does it perform in \(3 \times 10^2\) seconds?
  3. The Sun is approximately \(1.5 \times 10^{11}\) meters from Earth. Light travels at \(3.0 \times 10^8\) meters per second. How many seconds does it take light to travel from the Sun to Earth?
  4. The observable universe is estimated to be \(8.8 \times 10^{26}\) meters across. Using the speed of light above, approximately how many seconds would it take light to travel that distance?
  5. A gym membership has \(4.8 \times 10^3\) members. Each member pays \(\$3.6 \times 10^2\) per year. What is the gym's total annual revenue? Write your answer in scientific notation and in standard form.
  6. A chemist has a solution containing \(6.022 \times 10^{23}\) molecules per mole. She uses \(2.0 \times 10^{-3}\) moles in an experiment. How many molecules does she use?

Error Analysis

Each problem contains an error. Identify the error, explain what went wrong, and write the correct solution.

Problem 1
0.00042 = 4.2 × 10^4

What is the error? What is the correct answer?

Problem 2
(3 × 10^5) × (4 × 10^3) = 12 × 10^15

What is the error? What is the correct answer in proper scientific notation?

Problem 3
0.5 × 10^6 corrected to 5 × 10^7

What is the error in the correction? What is the correct proper form?

Problem 4
6.2 × 10^{-4} written in standard form as 62,000

What is the error? What is the correct standard form?

Problem 5
(8 × 10^5) ÷ (4 × 10^2) = 2 × 10^{2.5}

What is the error? What is the correct answer?

Problem 6
To enter 3.7 × 10^{-5} on a calculator, a student presses: 3.7 × 10 − 5

What is the error? How should the number be entered correctly?

Chapter 8 Checkpoint

Chapter 8 Checkpoint
Complete without assistance — assesses all key skills and concepts from Chapter 8
Part A — Vocabulary (5 points)

Match each term with its definition.

TermDefinition
1. Scientific notationA. The way a calculator displays scientific notation using the letter E
2. CoefficientB. The number between 1 and 10 in a scientific notation expression
3. Order of magnitudeC. A regular decimal number written without exponents
4. Standard formD. The power of ten that describes the scale of a number
5. E notationE. The compact form \(a \times 10^n\) where \(1 \leq a < 10\)
Part B — Powers of Ten (10 points)
  1. Evaluate \(10^5\) in standard form.
  2. Evaluate \(10^{-3}\) in standard form.
  3. Evaluate \(10^0\) in standard form.
  4. Evaluate \(10^{10}\) in standard form.
  5. Evaluate \(10^{-7}\) in standard form.
Part C — Conversion (20 points)

Write in scientific notation.

  1. 860,000
  2. 0.0000034
  3. 72,000,000,000
  4. 0.000000009
  5. 450

Write in standard form.

  1. \(5 \times 10^6\)
  2. \(3.1 \times 10^{-4}\)
  3. \(7.02 \times 10^8\)
  4. \(9.0 \times 10^{-11}\)
  5. \(1.5 \times 10^3\)
Part D — Operations (20 points)

Compute. Write each answer in proper scientific notation.

  1. \((4 \times 10^3) \times (3 \times 10^5)\)
  2. \((2 \times 10^{-2}) \times (5 \times 10^6)\)
  3. \(\dfrac{9 \times 10^8}{3 \times 10^5}\)
  4. \(\dfrac{6 \times 10^{-3}}{2 \times 10^4}\)
  5. \(\dfrac{(2 \times 10^6) \times (4 \times 10^3)}{8 \times 10^4}\)
Part E — Estimation (10 points)
  1. Estimate \((3.2 \times 10^7) \times (4.8 \times 10^3)\) to one significant figure.
  2. Estimate \((9.1 \times 10^{12}) \div (2.9 \times 10^4)\) to one significant figure.
Part F — Applications (15 points)
  1. A strand of spider silk is approximately \(3 \times 10^{-6}\) meters in diameter. A human hair is approximately \(7 \times 10^{-5}\) meters in diameter. How many times wider is a human hair than a strand of spider silk?
  2. A computer processor performs \(4 \times 10^9\) operations per second. How many operations does it perform in \(3 \times 10^2\) seconds? Write your answer in scientific notation.
  3. The Sun is approximately \(1.5 \times 10^{11}\) meters from Earth. Light travels at \(3.0 \times 10^8\) meters per second. How many seconds does it take light to travel from the Sun to Earth?
Part G — Error Analysis (10 points)
  1. A student writes that \(53 \times 10^4 = 5.3 \times 10^4\). Explain the error and give the correct answer.
  2. A student computes \((5 \times 10^3) \div (2 \times 10^7)\) and writes \(2.5 \times 10^4\). Identify the error and give the correct answer.

Answers for Chapter 8 may be placed in the answer key.

Bridge Note to Algebra I and Science
Chapter 9: Variables and Expressions — Beginning of Part IV: Pre-Algebra Foundations

Scientific notation will appear throughout your future courses. In Algebra I, you will use it when evaluating expressions with large or small values and when solving equations that arise from science and engineering contexts. In chemistry, Avogadro's number (\(6.022 \times 10^{23}\)) and molar mass calculations depend on fluent use of scientific notation. In physics, the gravitational constant (\(6.674 \times 10^{-11}\)), Planck's constant (\(6.626 \times 10^{-34}\)), and the speed of light (\(3.0 \times 10^8\)) are all expressed in this form. In earth science and astronomy, all distance and mass measurements at cosmic scales require it.

Chapter 9 opens Part IV: Pre-Algebra Foundations. There you will meet variables, the symbolic heart of algebra. The number sense and notation fluency you have built through Chapter 8 will carry forward directly into writing and evaluating algebraic expressions.