MGU Mathematics Series — Algebra II
Part III: Rational, Radical, and Power Functions
Roots · Rational Exponents · Radical Functions · Equations · Models
Chapter 8 of 14
MGU Mathematics Series  ·  Algebra II  ·  Library Textbook Edition

Chapter 8:
Radical Functions
and Rational Exponents

nth roots, rational exponents, radical simplification, operations, graphing, transformations, and solving radical equations — two notations, one mathematical idea

■ Math Level: Algebra II ■

■ 14 Sections  •  Part III of 5 ■

Follows Chapter 7: Rational Expressions and Functions  ·  Prepares for Chapter 9: Exponential Functions

Chapter 8 — Front Matter

Purpose and Overview

Chapter 8 extends the idea of radicals from expressions into functions, graphs, equations, and models. In Algebra I, you learned to simplify square roots, work with cube roots, and apply radicals to geometry through the Pythagorean theorem. In this chapter, those ideas grow into a broader framework that connects roots, powers, and inverse relationships.

The central insight of this chapter is that radicals and rational exponents are two notations for the same mathematical idea. A square root is a one-half power. A cube root is a one-third power. Recognizing this equivalence allows you to apply exponent rules to radicals, graph radical functions as transformations, solve equations involving roots, and interpret radical models in science, geometry, and engineering.

This chapter also introduces a critical habit: checking for extraneous solutions. When solving radical equations, the algebraic process can introduce false answers that must be identified and rejected. This kind of careful verification is not just a technique for radical equations — it reflects the broader mathematical discipline of asking whether an answer actually satisfies the original conditions.

Opening Question

A structural engineer is designing a cable support for a suspension bridge. The natural frequency at which the cable vibrates depends on the tension in the cable and the cable's mass per unit length. The formula involves a square root. The engineer needs to know: if the tension in a cable is quadrupled, what happens to the frequency? To answer this question, you need to understand how radical functions behave, how they respond to changes in their inputs, and how to reason about their outputs without computing every value numerically. By the end of this chapter, you will have exactly those tools.

14Sections
12Objectives
17Key Terms
20+Worked Examples

Learning Objectives

  1. 1Review and extend the meaning of square roots and cube roots from Algebra I.
  2. 2Define and evaluate nth roots for any positive integer n.
  3. 3Write and interpret rational exponents in the form \(a^{m/n}\).
  4. 4Convert fluently between radical notation and rational exponent notation.
  5. 5Simplify radical expressions using product and quotient properties.
  6. 6Add, subtract, multiply, and divide radical expressions.
  7. 7Define radical functions and state their domains.
  8. 8Graph square root and cube root functions from tables and formulas.
  9. 9Apply transformations — shifts, reflections, and stretches — to radical functions.
  10. 10Solve equations involving radicals by isolating the radical and raising both sides to a power.
  11. 11Identify and reject extraneous solutions produced during the solving process.
  12. 12Apply radical functions and equations to geometric and scientific models.
Chapter 8 — Contents

Sections at a Glance

Sections 8.1 – 8.4
Roots, Rational Exponents, and Two Notations for One Idea
Reviewing Radicals · nth Roots · Rational Exponents · Converting Forms
Section 8.1

Reviewing Radicals

What a Radical Expression Is

A radical expression is any expression that contains a root symbol, called the radical sign: \(\sqrt{\phantom{x}}\). The number or expression under the radical sign is called the radicand. The small number written above and to the left of the radical sign is called the index, indicating which root is being taken.

When no index is written, the index is understood to be 2, meaning a square root.

\[\sqrt[n]{a}\]

In this expression, \(n\) is the index and \(a\) is the radicand.

Square Roots

The square root of a number \(a\) is the number whose square equals \(a\): \(\sqrt{a} = b\) means \(b^2 = a\). Every positive real number has two square roots — one positive and one negative — but the symbol \(\sqrt{a}\) refers specifically to the principal square root, meaning the non-negative square root.

\[\sqrt{25} = 5 \qquad \text{(not } -5\text{)}\]
Student Warning

The expression \(-\sqrt{81}\) is not the same as \(\sqrt{-81}\). The first applies a negative sign to the principal square root of 81, giving \(-9\). The second asks for the square root of a negative number, which is not a real number.

Cube Roots

The cube root of a number \(a\) is the number whose cube equals \(a\). Unlike square roots, cube roots are defined for all real numbers, including negatives, because \((-2)^3 = -8\).

\[\sqrt[3]{-8} = -2\]
Key Distinction

Even-indexed radicals (square root, fourth root, …) require non-negative radicands for a real result. Odd-indexed radicals (cube root, fifth root, …) accept any real number as the radicand.

Example 8.1.1
Evaluating Square and Cube Roots
  • ProblemEvaluate (a) \(\sqrt{144}\)   (b) \(\sqrt[3]{27}\)   (c) \(-\sqrt{81}\)   (d) \(\sqrt[3]{-125}\)
  • ThinkIdentify the root and check whether the radicand is a perfect square or cube. Apply the definition directly.
  • Solve (a) \(\sqrt{144} = 12\), because \(12^2 = 144\).
    (b) \(\sqrt[3]{27} = 3\), because \(3^3 = 27\).
    (c) \(-\sqrt{81} = -9\). The negative sign is outside the radical; the principal root is 9, then the sign is applied.
    (d) \(\sqrt[3]{-125} = -5\), because \((-5)^3 = -125\).
  • CheckVerify by squaring or cubing each result.
Answer: 12  ·  3  ·  −9  ·  −5
Section 8.2

nth Roots

The nth root of a number \(a\), written \(\sqrt[n]{a}\), is the number \(b\) such that \(b^n = a\). This extends naturally to any positive integer index greater than 1.

\[\sqrt[n]{a} = b \quad \text{means} \quad b^n = a\]
ExpressionReads asResultCheck
\(\sqrt[4]{16}\)fourth root of 162\(2^4 = 16\) ✓
\(\sqrt[5]{32}\)fifth root of 322\(2^5 = 32\) ✓
\(\sqrt[4]{81}\)fourth root of 813\(3^4 = 81\) ✓
\(\sqrt[6]{64}\)sixth root of 642\(2^6 = 64\) ✓
Even vs. Odd Index — The Essential Distinction

Even index — Radicand must be non-negative. \(\sqrt[n]{-a}\) is not real. Principal root is non-negative. \(\sqrt[n]{a^n} = |a|\).

Odd index — Radicand can be any real number. Negative radicands give negative roots. \(\sqrt[n]{a^n} = a\) for all real \(a\).

Example 8.2.1
Evaluating nth Roots
  • ProblemEvaluate each expression, if it is real: (a) \(\sqrt[4]{625}\)   (b) \(\sqrt[5]{-32}\)   (c) \(\sqrt[4]{-16}\)
  • ThinkCheck whether the index is even or odd, and whether the radicand allows a real result.
  • Solve (a) \(\sqrt[4]{625} = 5\), because \(5^4 = 625\).
    (b) \(\sqrt[5]{-32} = -2\), because \((-2)^5 = -32\). Odd index permits a negative radicand.
    (c) \(\sqrt[4]{-16}\) is not a real number. The index is even and the radicand is negative.
Answer: 5  ·  −2  ·  not a real number
Sections 8.3 – 8.4
Rational Exponents and Converting Forms
Defining \(a^{1/n}\) and \(a^{m/n}\) · Exponent Rules Extended · Two-Way Conversion
Section 8.3

Rational Exponents

Defining \(a^{1/n}\)

Apply the power rule: if \(\left(8^{1/3}\right)^3 = 8^{(1/3)\cdot 3} = 8^1 = 8\), then \(8^{1/3}\) is a number whose cube equals 8. That is exactly the definition of a cube root. Therefore:

\[a^{1/n} = \sqrt[n]{a}\]

Defining \(a^{m/n}\)

The rational exponent \(m/n\) combines a root and a power. The denominator is the root index; the numerator is the power. There are two equivalent interpretations:

\[a^{m/n} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}\]
Practical Tip — Root First

In practice, take the root first, then apply the power. This keeps numbers smaller and reduces arithmetic errors. For example, \(64^{3/2}\): take \(\sqrt{64} = 8\), then cube: \(8^3 = 512\). Much easier than squaring 64 first.

Example 8.3.1
Evaluating Rational Exponents
  • ProblemEvaluate (a) \(27^{1/3}\)   (b) \(64^{3/2}\)   (c) \(16^{-3/4}\)
  • ThinkIdentify root (denominator) and power (numerator). Take the root first.
  • Solve (a) \(27^{1/3} = \sqrt[3]{27} = 3\)

    (b) \(64^{3/2} = \left(\sqrt{64}\right)^3 = 8^3 = 512\)

    (c) \(16^{-3/4} = \dfrac{1}{16^{3/4}} = \dfrac{1}{\left(\sqrt[4]{16}\right)^3} = \dfrac{1}{2^3} = \dfrac{1}{8}\)
  • Check(a) \(3^3 = 27\) ✓   (b) \(512^{2/3} = (\sqrt[3]{512})^2 = 8^2 = 64\) ✓   (c) \((1/8)^{-4/3} = 8^{4/3} = 2^4 = 16\) ✓
Answer: 3  ·  512  ·  1/8
Example 8.3.2
Using Exponent Rules with Rational Exponents
  • ProblemSimplify with positive rational exponents: (a) \(x^{1/2} \cdot x^{1/3}\)   (b) \(\dfrac{y^{3/4}}{y^{1/4}}\)   (c) \(\left(z^{2/3}\right)^{3/2}\)
  • ThinkApply standard exponent rules — add, subtract, or multiply the fractional exponents.
  • Solve (a) \(x^{1/2} \cdot x^{1/3} = x^{3/6 + 2/6} = x^{5/6}\)

    (b) \(\dfrac{y^{3/4}}{y^{1/4}} = y^{3/4 - 1/4} = y^{2/4} = y^{1/2}\)

    (c) \(\left(z^{2/3}\right)^{3/2} = z^{(2/3)(3/2)} = z^{6/6} = z\)
Answer: \(x^{5/6}\)  ·  \(y^{1/2}\)  ·  \(z\)
Section 8.4

Converting Between Radical and Exponential Form

Conversion Rules

\(\sqrt[n]{a^m} = a^{m/n}\)    The index becomes the denominator; the power becomes the numerator.

\(a^{m/n} = \sqrt[n]{a^m}\)    The denominator becomes the index; the numerator becomes the power.

Example 8.4.1
Converting from Radical to Exponential Form
  • ProblemWrite using a rational exponent: (a) \(\sqrt[5]{x^3}\)   (b) \(\sqrt{y^7}\)   (c) \(\sqrt[4]{a^3 b^5}\)
  • Solve (a) \(\sqrt[5]{x^3} = x^{3/5}\)
    (b) \(\sqrt{y^7} = y^{7/2}\)
    (c) \(\sqrt[4]{a^3 b^5} = a^{3/4} \cdot b^{5/4}\)
Answer: \(x^{3/5}\)  ·  \(y^{7/2}\)  ·  \(a^{3/4}b^{5/4}\)
Example 8.4.2
Converting from Exponential to Radical Form
  • ProblemWrite using a radical: (a) \(x^{2/7}\)   (b) \((3a)^{3/4}\)   (c) \(5^{1/2}\)
  • Solve (a) \(x^{2/7} = \sqrt[7]{x^2}\)
    (b) \((3a)^{3/4} = \sqrt[4]{(3a)^3} = \sqrt[4]{27a^3}\)
    (c) \(5^{1/2} = \sqrt{5}\)
Answer: \(\sqrt[7]{x^2}\)  ·  \(\sqrt[4]{27a^3}\)  ·  \(\sqrt{5}\)
Student Warning

When converting \((3a)^{3/4}\), the exponent applies to the entire product \(3a\). Write the entire expression under the radical before simplifying: \(\sqrt[4]{(3a)^3}\), not \(\sqrt[4]{3} \cdot a^3\).

Practice 8.1–8.4Rational Exponents and Conversion
  1. Evaluate \(25^{1/2}\)
  2. Evaluate \(8^{2/3}\)
  3. Evaluate \(16^{3/4}\)
  4. Evaluate \(27^{-1/3}\)
  5. Evaluate \(32^{3/5}\)
  6. Write \(\sqrt[3]{x^4}\) as \(x^{m/n}\)
  7. Write \(\sqrt[6]{y^5}\) as \(y^{m/n}\)
  8. Write \(x^{3/8}\) as a radical
  9. Write \((2y)^{5/3}\) as a radical
  10. Simplify \(x^{2/3} \cdot x^{1/6}\)
  11. Simplify \(\dfrac{y^{5/4}}{y^{3/4}}\)
  12. Simplify \(\left(z^{3/5}\right)^{10/3}\)
Sections 8.5 – 8.6
Simplifying and Operations with Radicals
Product & Quotient Properties · Rationalizing · Like Radicals · Multiplying · Conjugates
Section 8.5

Simplifying Radical Expressions

Product and Quotient Properties of Radicals

\(\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}\)    The radical of a product equals the product of the radicals.

\(\sqrt[n]{\dfrac{a}{b}} = \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}\)    The radical of a quotient equals the quotient of the radicals. \((b \neq 0)\)

Strategy: factor the radicand to expose perfect nth power factors, then extract them.

Example 8.5.1
Simplifying Square Roots
  • ProblemSimplify (a) \(\sqrt{72}\)   (b) \(\sqrt{180}\)   (c) \(\sqrt{50x^4 y^3}\)
  • ThinkFactor each radicand to find the largest perfect square factor. Apply the product rule.
  • Solve (a) \(\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}\)

    (b) \(\sqrt{180} = \sqrt{36 \cdot 5} = 6\sqrt{5}\)

    (c) \(\sqrt{50x^4 y^3} = \sqrt{25 \cdot 2 \cdot x^4 \cdot y^2 \cdot y} = 5x^2 y\sqrt{2y}\)
  • Check(c) \((5x^2y)^2 \cdot 2y = 25x^4y^2 \cdot 2y = 50x^4y^3\) ✓
Answer: \(6\sqrt{2}\)  ·  \(6\sqrt{5}\)  ·  \(5x^2 y\sqrt{2y}\)
Example 8.5.2
Simplifying Cube Roots
  • ProblemSimplify (a) \(\sqrt[3]{54}\)   (b) \(\sqrt[3]{16x^5}\)
  • Solve (a) \(\sqrt[3]{54} = \sqrt[3]{27 \cdot 2} = 3\sqrt[3]{2}\)

    (b) \(\sqrt[3]{16x^5} = \sqrt[3]{8 \cdot 2 \cdot x^3 \cdot x^2} = 2x\sqrt[3]{2x^2}\)
  • Check(b) \((2x)^3 \cdot 2x^2 = 8x^3 \cdot 2x^2 = 16x^5\) ✓
Answer: \(3\sqrt[3]{2}\)  ·  \(2x\sqrt[3]{2x^2}\)
Example 8.5.3
Rationalizing the Denominator
  • ProblemRationalize the denominator of \(\dfrac{6}{\sqrt{3}}\)
  • ThinkMultiply numerator and denominator by \(\sqrt{3}\) to eliminate the radical below.
  • Solve \[\frac{6}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}\]
Answer: \(2\sqrt{3}\)
Section 8.6

Operations with Radicals

Adding and Subtracting — Like Radicals

Radical expressions can be added or subtracted only when they have the same index and the same radicand. Such expressions are called like radicals. Combine them by adding or subtracting their coefficients. If expressions don't appear to match, simplify first — hidden like radicals often appear after simplification.

Example 8.6.1
Adding and Subtracting Radicals
  • ProblemSimplify (a) \(5\sqrt{3} + 2\sqrt{3} - \sqrt{3}\)   (b) \(\sqrt{50} + \sqrt{18}\)   (c) \(3\sqrt[3]{16} - \sqrt[3]{54}\)
  • ThinkFor (a), combine like radicals directly. For (b) and (c), simplify first to reveal like radicals.
  • Solve (a) \((5+2-1)\sqrt{3} = 6\sqrt{3}\)

    (b) \(\sqrt{50}+\sqrt{18} = 5\sqrt{2} + 3\sqrt{2} = 8\sqrt{2}\)

    (c) \(3\sqrt[3]{8\cdot2} - \sqrt[3]{27\cdot2} = 6\sqrt[3]{2} - 3\sqrt[3]{2} = 3\sqrt[3]{2}\)
Answer: \(6\sqrt{3}\)  ·  \(8\sqrt{2}\)  ·  \(3\sqrt[3]{2}\)
Student Warning

\(\sqrt{2} + \sqrt{3}\) cannot be combined. The radicands differ, so the expression is already in simplest form. Do not write \(\sqrt{5}\).

Multiplying Radicals

To multiply radicals with the same index, multiply the radicands. Multiply coefficients separately from radicands. Binomial radical products use the distributive property or FOIL.

Example 8.6.2
Multiplying Radical Expressions
  • ProblemExpand and simplify (a) \(\sqrt{3}(\sqrt{6}+4)\)   (b) \((\sqrt{5}+2)(\sqrt{5}-3)\)   (c) \((2\sqrt{3}+\sqrt{7})^2\)
  • ThinkDistribute or FOIL. Remember \(\sqrt{a}\cdot\sqrt{a} = a\).
  • Solve (a) \(\sqrt{3}(\sqrt{6}+4) = \sqrt{18}+4\sqrt{3} = 3\sqrt{2}+4\sqrt{3}\)

    (b) \((\sqrt{5}+2)(\sqrt{5}-3) = 5 - 3\sqrt{5} + 2\sqrt{5} - 6 = -1-\sqrt{5}\)

    (c) \((2\sqrt{3})^2 + 2(2\sqrt{3})(\sqrt{7}) + (\sqrt{7})^2 = 12 + 4\sqrt{21} + 7 = 19 + 4\sqrt{21}\)
Answer: \(3\sqrt{2}+4\sqrt{3}\)  ·  \(-1-\sqrt{5}\)  ·  \(19+4\sqrt{21}\)

Conjugate Pairs and Rationalizing Binomial Denominators

The expressions \(a+b\) and \(a-b\) are conjugates. Their product eliminates the middle terms: \((a+b)(a-b) = a^2 - b^2\). When one term is a square root, this product is rational. Use the conjugate to rationalize binomial radical denominators.

Example 8.6.3
Rationalizing a Binomial Denominator
  • ProblemRationalize: \(\dfrac{4}{\sqrt{7}-1}\)
  • ThinkMultiply by the conjugate \(\sqrt{7}+1\).
  • Solve \[\frac{4}{\sqrt{7}-1} \cdot \frac{\sqrt{7}+1}{\sqrt{7}+1} = \frac{4(\sqrt{7}+1)}{7-1} = \frac{4\sqrt{7}+4}{6} = \frac{2\sqrt{7}+2}{3}\]
Answer: \(\dfrac{2\sqrt{7}+2}{3}\)
Practice 8.5–8.6Simplifying and Operations
  1. \(\sqrt{200}\)
  2. \(\sqrt{98x^3}\)
  3. \(\sqrt[3]{108}\)
  4. \(\sqrt[4]{48x^8}\)
  5. \(5\sqrt{3}+2\sqrt{12}\)
  6. \(\sqrt{75}-3\sqrt{27}\)
  7. \(\sqrt{6}\cdot\sqrt{10}\)
  8. \((3\sqrt{5})(2\sqrt{15})\)
  9. \((\sqrt{3}+4)(\sqrt{3}-4)\)
  10. \((2\sqrt{5}-1)^2\)
  11. \(\dfrac{10}{\sqrt{5}}\)
  12. \(\dfrac{6}{\sqrt{7}+1}\)
Sections 8.7 – 8.10
Radical Functions — Domains, Graphs, and Transformations
Defining Radical Functions · Square Root Function · Cube Root Function · Transformations
Section 8.7

Radical Functions

A radical function is a function defined by a rule that contains a radical expression with a variable under the radical sign. The domain — the set of all allowable inputs — depends on the index.

For even-indexed radical functions, set the radicand \(\geq 0\) and solve the inequality. For odd-indexed radical functions, the domain is all real numbers.

Example 8.7.1
Finding the Domain of Radical Functions
  • ProblemState the domain of (a) \(f(x)=\sqrt{x-3}\)   (b) \(g(x)=\sqrt{2x+8}\)   (c) \(h(x)=\sqrt[3]{x+5}\)
  • ThinkEven roots: radicand ≥ 0. Odd roots: all reals.
  • Solve (a) \(x-3 \geq 0 \Rightarrow x \geq 3\). Domain: \([3,\infty)\)
    (b) \(2x+8 \geq 0 \Rightarrow x \geq -4\). Domain: \([-4,\infty)\)
    (c) Odd index; domain: \((-\infty,\infty)\)
Answer: \([3,\infty)\)  ·  \([-4,\infty)\)  ·  \((-\infty,\infty)\)
Inverse Relationships

Radical functions are inverses of power functions. The function \(f(x)=\sqrt{x}\) undoes squaring. The function \(f(x)=\sqrt[3]{x}\) undoes cubing. This inverse relationship will deepen in Chapters 9 and 10 with exponential and logarithmic functions.

Section 8.8

Graphing Square Root Functions

The Parent Square Root Function

The parent square root function is \(f(x) = \sqrt{x}\). Its domain is \(x \geq 0\) and its range is \(y \geq 0\). The graph begins at the origin \((0,0)\) and curves upward and to the right, growing more slowly as \(x\) increases.

\(x\)\(f(x) = \sqrt{x}\)
00
11
42
93
164
255
Diagram Instruction

Draw a coordinate plane. Plot the points (0,0), (1,1), (4,2), (9,3), and (16,4). Connect them with a smooth curve that begins at the origin and curves upward and to the right, concave down (the rate of growth decreases as x increases). Label the curve \(f(x)=\sqrt{x}\). Shade the domain along the x-axis to indicate x ≥ 0.

Key Characteristics — \(f(x)=\sqrt{x}\)

Domain: \([0,\infty)\)   Range: \([0,\infty)\)

Intercepts: Both x- and y-intercept at (0,0)

Behavior: Increasing throughout; rate of growth decreases as x grows

End behavior: As \(x \to \infty\), \(f(x) \to \infty\) slowly

Example 8.8.1
Graphing a Square Root Function
  • ProblemGraph \(f(x) = \sqrt{x+4}\) and state its domain and range.
  • ThinkThis shifts the parent \(\sqrt{x}\) four units to the left. The graph starts where \(x+4=0\), that is, at \(x=-4\).
  • Solve
    \(x\)\(x+4\)\(f(x)\)
    −400
    −311
    042
    593
    12164
Domain: \([-4,\infty)\)  ·  Range: \([0,\infty)\)
Diagram Instruction

Draw a coordinate plane. Plot points (−4,0), (−3,1), (0,2), (5,3), and (12,4). Connect with a smooth curve starting at (−4,0) and curving up and to the right. Draw the parent function \(\sqrt{x}\) as a dashed curve for comparison. Label the solid curve \(f(x)=\sqrt{x+4}\) and note the 4-unit left shift.

Section 8.9

Graphing Cube Root Functions

The Parent Cube Root Function

The parent cube root function is \(f(x) = \sqrt[3]{x}\). Unlike the square root function, it is defined for all real numbers. Its domain and range are both \((-\infty,\infty)\). The graph passes through the origin in an S-shaped curve that extends into the third and first quadrants.

\(x\)\(f(x)=\sqrt[3]{x}\)
−8−2
−1−1
00
11
82
273
Diagram Instruction

Draw a coordinate plane. Plot (−8,−2), (−1,−1), (0,0), (1,1), (8,2), and (27,3). Connect with a smooth S-shaped curve passing through the origin. The curve increases throughout: steep near the origin, flattening as |x| grows. It extends into both the third and first quadrants. Label the curve \(f(x)=\sqrt[3]{x}\).

Key Characteristics — \(f(x)=\sqrt[3]{x}\)

Domain: \((-\infty,\infty)\)   Range: \((-\infty,\infty)\)

Intercepts: Both at origin (0,0)

Symmetry: Point symmetry about the origin — odd function

Behavior: Increasing throughout; S-shaped curve

Example 8.9.1
Graphing a Cube Root Function
  • ProblemGraph \(g(x) = \sqrt[3]{x-2}+1\) and state its domain and range.
  • ThinkShifts the parent 2 units right and 1 unit up. Center of the S-curve moves from (0,0) to (2,1).
  • Solve
    \(x\)\(x-2\)\(\sqrt[3]{x-2}\)\(g(x)\)
    −6−8−2−1
    1−1−10
    2001
    3112
    10823
Domain: \((-\infty,\infty)\)  ·  Range: \((-\infty,\infty)\)
Section 8.10

Transformations of Radical Functions

General Transformation Form

\(f(x) = a\sqrt[n]{x-h}+k\)

a — Vertical stretch (\(|a|>1\)) or compression (\(0<|a|<1\)); if \(a<0\), reflects over the x-axis.

h — Horizontal shift: graph moves h units right (left if \(h<0\)).

k — Vertical shift: graph moves k units up (down if \(k<0\)).

For even-indexed radicals, the starting point shifts to \((h,k)\). For odd-indexed radicals, the center/inflection point shifts to \((h,k)\).

Example 8.10.1
Identifying Transformations
  • ProblemDescribe transformations and state domain and range for (a) \(f(x)=-\sqrt{x}+3\)   (b) \(g(x)=2\sqrt{x+1}-4\)   (c) \(h(x)=-\sqrt[3]{x-5}\)
  • Solve (a) \(a=-1\), \(h=0\), \(k=3\): reflected over x-axis, shifted 3 up. Domain: \([0,\infty)\). Range: \((-\infty,3]\).

    (b) \(a=2\), \(h=-1\), \(k=-4\): stretched by factor 2, shifted 1 left and 4 down. Domain: \([-1,\infty)\). Range: \([-4,\infty)\).

    (c) \(a=-1\), \(h=5\), \(k=0\): reflected over x-axis, shifted 5 right. Domain: \((-\infty,\infty)\). Range: \((-\infty,\infty)\).
Transformations and domains/ranges as described above.
Practice 8.7–8.10Radical Functions and Graphs

For each function: state the domain and range, identify transformations from the parent, and describe the graph.

  1. \(f(x) = \sqrt{x-2}\)
  2. \(g(x) = -\sqrt{x}+5\)
  3. \(h(x) = 3\sqrt{x+1}-2\)
  4. \(f(x) = \sqrt[3]{x+4}\)
  5. \(g(x) = -2\sqrt[3]{x-1}+3\)
  6. \(h(x) = \sqrt[3]{-x}\)
  7. \(f(x) = \sqrt{4x-12}\)
  8. \(g(x) = \frac{1}{2}\sqrt{x+3}+1\)
Sections 8.11 – 8.12
Solving Radical Equations and Extraneous Solutions
Isolate · Raise to a Power · Solve · Always Check
Section 8.11

Solving Radical Equations

General Procedure for Solving Radical Equations

1. Isolate the radical on one side of the equation.

2. Raise both sides to the power equal to the index of the radical.

3. Solve the resulting equation (linear, quadratic, or other).

4. Check every solution in the original equation. Reject extraneous solutions.

Example 8.11.1
Solving a Basic Radical Equation
  • ProblemSolve \(\sqrt{x+3}=7\)
  • ThinkThe radical is already isolated. Square both sides.
  • Solve \[\left(\sqrt{x+3}\right)^2 = 7^2 \implies x+3=49 \implies x=46\]
  • Check\(\sqrt{46+3} = \sqrt{49} = 7\) ✓
Answer: \(x = 46\)
Example 8.11.2
Isolating Before Squaring
  • ProblemSolve \(\sqrt{3x-2}-4=0\)
  • Solve \[\sqrt{3x-2}=4 \implies 3x-2=16 \implies x=6\]
  • Check\(\sqrt{3(6)-2}-4 = \sqrt{16}-4 = 0\) ✓
Answer: \(x = 6\)
Example 8.11.3
Solving a Cube Root Equation
  • ProblemSolve \(\sqrt[3]{2x-1}=4\)
  • ThinkCube both sides to eliminate the cube root. Odd-index equations do not produce extraneous solutions, but checking is good practice.
  • Solve \[2x-1=4^3=64 \implies 2x=65 \implies x=\frac{65}{2}\]
  • Check\(\sqrt[3]{2 \cdot \frac{65}{2}-1} = \sqrt[3]{64} = 4\) ✓
Answer: \(x = \dfrac{65}{2}\)
Example 8.11.4
Radical Equation Leading to a Quadratic
  • ProblemSolve \(\sqrt{x+5}=x-1\)
  • ThinkSquare both sides. This produces a quadratic with two possible solutions — check both.
  • Solve \[x+5 = (x-1)^2 = x^2-2x+1\] \[0 = x^2-3x-4 = (x-4)(x+1)\] \[x=4 \quad \text{or} \quad x=-1\]
  • Check \(x=4\): \(\sqrt{9}=3\) and \(4-1=3\) ✓
    \(x=-1\): \(\sqrt{4}=2\) and \(-1-1=-2\) ✗   Extraneous!
Answer: \(x = 4\) only. (\(x=-1\) is extraneous.)
Example 8.11.5
Equation with Two Radicals
  • ProblemSolve \(\sqrt{x+7}=\sqrt{x}+1\)
  • ThinkSquare both sides; the right side expands as a binomial. A radical will remain — isolate it and square again.
  • Solve \[x+7 = (\sqrt{x}+1)^2 = x+2\sqrt{x}+1\] \[6 = 2\sqrt{x} \implies \sqrt{x}=3 \implies x=9\]
  • Check\(\sqrt{16}=4\) and \(\sqrt{9}+1=4\) ✓
Answer: \(x = 9\)
Section 8.12

Extraneous Solutions

An extraneous solution is a value that satisfies an equation obtained by algebraic manipulation but does not satisfy the original equation. Extraneous solutions arise in radical equations because squaring both sides is not a reversible operation.

If \(A=B\), then \(A^2=B^2\). But \(A^2=B^2\) does not guarantee \(A=B\) — it also allows \(A=-B\). Squaring can introduce false solutions from the case where one side was actually the negative of the other.

Example 8.12.1
No Valid Solution
  • ProblemSolve \(\sqrt{x-4}+8=5\)
  • Solve\(\sqrt{x-4} = -3\)
  • CheckA principal square root is always non-negative. It cannot equal \(-3\). Stop here.
Answer: No solution. Solution set is \(\emptyset\).
Student Warning — Recognize This Pattern Immediately

If isolating the radical gives \(\sqrt{\text{something}} = \text{negative number}\), there is no real solution. Do not proceed to square both sides. Squaring would give a numerical answer, but that answer would be entirely extraneous.

Summary — Handling Extraneous Solutions

1. Solve to get all candidate solutions.

2. Substitute each candidate into the original equation (not the squared version).

3. Accept those that make the original true; reject those that do not.

4. If all candidates are rejected: no solution.

Practice 8.11–8.12Solving Radical Equations

Solve each equation. Check all solutions and identify any that are extraneous.

  1. \(\sqrt{x}=8\)
  2. \(\sqrt{x-5}=4\)
  3. \(\sqrt{3x+7}=5\)
  4. \(\sqrt{2x-3}-6=0\)
  5. \(\sqrt[3]{5x-2}=3\)
  6. \(\sqrt{x+6}=x\)
  7. \(\sqrt{4x+1}=x-1\)
  8. \(\sqrt{5x-4}=\sqrt{3x+8}\)
  9. \(\sqrt{x+10}-\sqrt{x}=2\)
  10. \(\sqrt{x-4}+8=5\)
Section 8.13
Radical Models in Geometry and Science
Pythagorean Theorem · Distance Formula · Pendulum Period · Engineering
Section 8.13

Radical Models in Geometry and Science

Radicals in Geometry

Key Geometric Formulas Involving Radicals

Pythagorean Theorem — For a right triangle with legs \(a\), \(b\) and hypotenuse \(c\): \(\;c = \sqrt{a^2+b^2}\)

Distance Formula — Between points \((x_1,y_1)\) and \((x_2,y_2)\): \(\;d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)

Square area/side — If area is \(A\): side length \(s = \sqrt{A}\)

Circle area/radius — If area is \(A\): radius \(r = \sqrt{A/\pi}\)

Example 8.13.1
Geometry — Diagonal of a Rectangular Park
  • ProblemA rectangular park is 150 m long and 80 m wide. How far is it diagonally across?
  • ThinkThe diagonal is the hypotenuse of a right triangle with legs 150 and 80.
  • Solve \[d = \sqrt{150^2+80^2} = \sqrt{22500+6400} = \sqrt{28900} = 170\]
  • Check\(170^2 = 28900 = 22500+6400\) ✓
Answer: The diagonal is 170 meters.

Radicals in Physics — The Pendulum

Period of a Simple Pendulum

The time \(T\) (in seconds) for one complete swing of a pendulum of length \(L\) (in meters):

\[T = 2\pi\sqrt{\frac{L}{g}}\]

where \(g \approx 9.8\) m/s² near Earth's surface. Because of the square root, quadrupling the length only doubles the period: \(\sqrt{4L}=2\sqrt{L}\).

Example 8.13.2
Pendulum Period
  • ProblemA clock pendulum is 0.25 m long. What is its period? Use \(g=9.8\) m/s², \(\pi \approx 3.14159\).
  • Solve \[T = 2\pi\sqrt{\frac{0.25}{9.8}} = 2(3.14159)\sqrt{0.02551} \approx 2(3.14159)(0.1597) \approx 1.00 \text{ sec}\]
  • CheckA 25 cm pendulum swinging once per second is exactly how many old mechanical clocks were calibrated. ✓
Answer: Period \(\approx\) 1.00 second.
Example 8.13.3
Solving a Science Formula for a Variable
  • ProblemA pendulum has a period of 2 seconds. Find its required length. Use \(g=9.8\) m/s².
  • ThinkSubstitute \(T=2\) and isolate \(L\) by isolating the radical, then squaring both sides.
  • Solve \[2 = 2\pi\sqrt{\frac{L}{9.8}} \implies 1=\pi\sqrt{\frac{L}{9.8}} \implies \frac{1}{\pi}=\sqrt{\frac{L}{9.8}}\] \[\frac{1}{\pi^2}=\frac{L}{9.8} \implies L=\frac{9.8}{\pi^2} \approx \frac{9.8}{9.8696} \approx 0.993 \text{ m}\]
Answer: Length \(\approx\) 0.993 meters (about 99.3 cm).
MGU Library Connection
  • The pendulum formula and wave-speed formulas appear in MGU Physics I and introductory engineering modules.
  • The distance formula appears throughout MGU Trigonometry, Precalculus, and Analytic Geometry chapters.
  • Standard deviation in MGU Statistics also involves a square root — connecting algebra to data analysis.
  • See Appendix H of this textbook: Radical and Rational Exponent Rules Reference Sheet.
Practice 8.13Applications
  1. A right triangle has legs of 9 cm and 40 cm. Find the hypotenuse.
  2. Points \(A=(2,-1)\) and \(B=(10,5)\) lie on a coordinate plane. Find the distance \(AB\).
  3. A square swimming pool has an area of 200 square feet. What is its side length, in simplest radical form?
  4. Use \(T=2\pi\sqrt{L/9.8}\). A pendulum has a period of 3 seconds. Find its length to the nearest hundredth of a meter.
  5. A 1.96-meter pendulum — what is its period? Use \(g=9.8\) m/s².
  6. An object dropped from height \(h\) feet hits the ground after \(t=\sqrt{h/16}\) seconds. From what height was it dropped if it fell for 4 seconds?
  7. A highway curve has a safe speed model \(v=\sqrt{2.5r}\) (mph, feet). For a safe max speed of 60 mph, what radius is needed?
  8. A bridge cable vibrates with frequency \(f=\frac{1}{2L}\sqrt{T/\mu}\). If tension is quadrupled and everything else is unchanged, what happens to the frequency? Explain using square root reasoning.
Section 8.14

Common Radical Function Mistakes

Mistake 1 — Distributing a Root Across a Sum
\(\sqrt{a+b} = \sqrt{a}+\sqrt{b}\)

Counterexample: \(\sqrt{9+16}=\sqrt{25}=5\), but \(\sqrt{9}+\sqrt{16}=3+4=7 \neq 5\). The radical of a sum cannot be split across the terms.

Mistake 2 — Forgetting to Check for Extraneous Solutions
Accepting every solution of the squared equation without checking in the original

Squaring is not reversible. Always substitute candidate solutions back into the original radical equation and reject those that fail.

Mistake 3 — Ignoring the Domain Restriction of Even-Indexed Radicals
Treating \(f(x)=\sqrt{x-5}\) as defined for all x

The domain is \(x \geq 5\). Even-indexed radical functions require non-negative radicands. Always determine the domain before graphing or solving.

Mistake 4 — Confusing \(-\sqrt{x}\) and \(\sqrt{-x}\)
\(-\sqrt{x} = \sqrt{-x}\)

\(-\sqrt{x}\) is the negative of the principal square root of \(x\) (defined for \(x \geq 0\)). The expression \(\sqrt{-x}\) requires \(x \leq 0\) and gives a non-negative output. These are different functions with different domains.

Mistake 5 — Dropping Absolute Values with Even-Indexed Roots
\(\sqrt{x^2}=x\) for all x

Correct: \(\sqrt{x^2}=|x|\). The principal square root is non-negative, so \(\sqrt{(-3)^2}=\sqrt{9}=3\), not \(-3\). For odd roots: \(\sqrt[3]{x^3}=x\) for all real \(x\).

Mistake 6 — Not Simplifying Before Combining
Concluding that \(\sqrt{50}+\sqrt{18}\) cannot be simplified

Simplify first: \(\sqrt{50}=5\sqrt{2}\), \(\sqrt{18}=3\sqrt{2}\), so their sum is \(8\sqrt{2}\). Like radicals often appear only after simplifying.

Mistake 7 — Inverting the Rational Exponent Rule
\(a^{m/n}=\sqrt[m]{a^n}\)

Correct: \(a^{m/n}=\sqrt[n]{a^m}\). The denominator is the index (root) and the numerator is the power. Swapping them gives a completely different — and wrong — expression.

Mistake 8 — Stopping After One Squaring When Two Are Needed
In \(\sqrt{x+7}=\sqrt{x}+1\), squaring once and then treating remaining radicals as numbers

After the first squaring, a radical may remain. Isolate it and square again. Continue until no radicals remain. Always check the final result.

Chapter 8 — Back Matter

Chapter Summary

Key Terms Review

radical expression
An expression containing a root symbol with a radicand under it and an index indicating which root is taken.
radicand
The expression placed under the radical sign.
index
The integer written above and to the left of the radical sign, indicating which root is being taken. Omitted index defaults to 2 (square root).
nth root
The nth root of a number a is the number b such that b raised to the nth power equals a.
principal root
The non-negative real nth root for even indices. The radical symbol always refers to the principal root.
rational exponent
An exponent written as a fraction m/n: the denominator is the root index and the numerator is the power. \(a^{m/n}=\sqrt[n]{a^m}\).
irrational number
A real number that cannot be expressed as a ratio of integers. Many simplified radical expressions, like \(\sqrt{2}\), are irrational.
like radicals
Radicals with the same index and the same radicand. Only like radicals can be combined by addition or subtraction.
conjugate radical pair
Expressions of the form \(a+b\) and \(a-b\) where at least one term contains a square root. Their product \(a^2-b^2\) contains no radicals.
radical function
A function whose rule contains a variable under a radical sign.
domain restriction
A limitation on allowable inputs. For even-indexed radical functions, the radicand must be non-negative.
radical equation
An equation in which the variable appears under a radical sign.
extraneous solution
A value that satisfies an equation obtained by algebraic steps but does not satisfy the original equation. Must be identified and rejected.
square root function
The parent function \(f(x)=\sqrt{x}\) with domain and range \([0,\infty)\). Starts at the origin and curves up and to the right.
cube root function
The parent function \(f(x)=\sqrt[3]{x}\) with domain and range \((-\infty,\infty)\). S-shaped curve through the origin.
transformation
A shift, stretch, compression, or reflection applied to a parent function via the parameters a, h, and k.
inverse function
A function that reverses the input-output relationship of another. Radical functions are inverses of power functions with appropriate domain restrictions.

Concept Review Questions

  1. Explain the difference between the principal square root and the negative square root of a number. Why does \(\sqrt{a}\) refer to the principal root only?
  2. Why do even-indexed radicals require non-negative radicands, while odd-indexed radicals do not? Connect your answer to the meaning of raising a number to an even vs. odd power.
  3. Describe the connection between \(a^{m/n}\) and \(\sqrt[n]{a^m}\). Which part of the rational exponent is the root, and which is the power?
  4. Why is it not always valid to write \(\sqrt[n]{a^n}=a\) when \(n\) is even? Give an example with a negative value of \(a\).
  5. Define like radicals and give two examples of like pairs and two examples of unlike pairs.
  6. What is the purpose of rationalizing a denominator? When does a conjugate pair help accomplish it?
  7. Compare the domain and range of \(f(x)=\sqrt{x}\) and \(f(x)=\sqrt[3]{x}\). Explain why they differ.
  8. Describe what \(a\), \(h\), and \(k\) each do in the function \(f(x)=a\sqrt[n]{x-h}+k\).
  9. Explain why extraneous solutions arise when solving radical equations. What algebraic step introduces them?
  10. Name one geometric formula and one physics formula that involves a radical. Describe what the radical represents in each context.

Skill Practice

Part ARational Exponents
  1. Evaluate \(81^{3/4}\)
  2. Evaluate \(32^{-2/5}\)
  3. Simplify \((16a^4)^{3/4}\)
  4. Simplify \(\dfrac{x^{5/3}}{x^{2/3}}\)
  5. Write \(\sqrt[5]{m^3n^2}\) with rational exponents
  6. Write \(b^{7/2}\) as a radical
Part BRadical Simplification
  1. \(\sqrt{98x^3}\)
  2. \(\sqrt[3]{-250y^5}\)
  3. \(\sqrt[4]{48x^8}\)
  4. \(\sqrt{\dfrac{x^5}{y^4}}\)
  5. \(\dfrac{\sqrt{32}}{\sqrt{2}}\)
  6. \(\sqrt{\dfrac{75}{49}}\)
Part COperations with Radicals
  1. \(4\sqrt[3]{2}+\sqrt[3]{54}\)
  2. \(\sqrt{48}+\sqrt{27}-\sqrt{12}\)
  3. \(\sqrt{2}(\sqrt{8}+3)\)
  4. \((2\sqrt{3}+\sqrt{7})^2\)
  5. \(\dfrac{\sqrt{3}}{\sqrt{6}-\sqrt{3}}\)
  6. \(\dfrac{3}{\sqrt[3]{4}}\)
Part DRadical Equations
  1. Solve \(\sqrt{x-5}=4\)
  2. Solve \(\sqrt{3x+7}=5\)
  3. Solve \(\sqrt[3]{5x-2}=3\)
  4. Solve \(\sqrt{x+6}=x\)
  5. Solve \(\sqrt{4x+1}=x-1\)
  6. Solve \(\sqrt{x+10}-\sqrt{x}=2\)

Word Problems

Define variables, write a model, and answer the specific question.

  1. A square garden has an area of 338 m². What is its side length? Write in simplest radical form and as a decimal to the nearest hundredth.
  2. Points \(A=(-3,1)\) and \(B=(5,7)\) lie on a coordinate plane. Find the distance \(AB\) in simplest radical form.
  3. Use \(T=2\pi\sqrt{L/9.8}\). A pendulum must have a period of 1.5 seconds. Find the required length to the nearest hundredth of a meter.
  4. An object falls from a building. The time in seconds to reach the ground is \(t=\sqrt{h/16}\), where h is height in feet. If the object falls for 5 seconds, from what height was it dropped?
  5. A highway curve model gives safe speed as \(v=\sqrt{2.5r}\) mph, where r is curve radius in feet. For 60 mph maximum, what radius is required?
  6. A finance formula gives the interest rate per period as \(r=\left(\dfrac{A}{P}\right)^{1/n}-1\). If \$1000 grows to \$1331 in 3 years with annual compounding, find the annual interest rate.

Error Analysis

Each problem shows a student error. Identify it and give the correct solution.

Problem 1
Simplify \(\sqrt{x^2+9}\): Student answer: \(x+3\)

What is wrong? What is the correct simplified form?

Problem 2
Solve \(\sqrt{x-2}=-4\): Student writes \(x-2=16\), so \(x=18\)

What is wrong? What should the student have concluded?

Problem 3
Simplify \(\sqrt{50}+\sqrt{8}\): Student answer: \(\sqrt{58}\)

What is wrong? Show the correct simplification.

Problem 4
Solve \(\sqrt{x+3}=x-3\): Student gets \(x=1\) or \(x=6\) and reports both as solutions without checking.

What critical step is missing? Which solutions are valid?

Problem 5
Write \(a^{2/3}\) as a radical: Student writes \(\sqrt[2]{a^3}\)

What is wrong? Write the correct radical form.

Chapter 8 Checkpoint

Chapter 8 Checkpoint
Complete without assistance — assesses all key skills and concepts from Chapter 8
Part 1 — Rational Exponents and Radical Notation
  1. Evaluate \(32^{4/5}\).
  2. Write \(\sqrt[4]{x^3 y^5}\) using rational exponents.
  3. Simplify \(x^{3/4} \cdot x^{5/4}\) and write the result in radical form.
  4. Write \((5z)^{2/3}\) as a radical expression.
Part 2 — Simplifying and Operating
  1. Simplify \(\sqrt{72x^5 y^4}\).
  2. Simplify \(\sqrt[3]{54a^4 b^7}\).
  3. Add: \(2\sqrt{45}+3\sqrt{20}-\sqrt{5}\).
  4. Multiply and simplify: \((\sqrt{6}+\sqrt{2})(\sqrt{6}-\sqrt{2})\).
  5. Rationalize the denominator: \(\dfrac{5}{3-\sqrt{2}}\).
Part 3 — Radical Functions and Graphs
  1. State the domain and range of \(f(x)=\sqrt{4x-12}\).
  2. Describe all transformations of \(g(x)=-3\sqrt{x+2}+5\) from the parent \(f(x)=\sqrt{x}\).
  3. Write the equation of a cube root function whose center is at \((-1,4)\) and that has been reflected over the x-axis.
Part 4 — Solving Radical Equations
  1. Solve \(\sqrt{5x-1}=7\). Check your answer.
  2. Solve \(\sqrt{x+8}=x-2\). Check for extraneous solutions.
  3. Solve \(\sqrt{3x+4}=\sqrt{x+10}\). Check your answer.
  4. Solve \(\sqrt{2x+7}+3=1\). Determine whether a solution exists.
Part 5 — Applications
  1. A square garden has an area of 338 m². Find the side length in simplest radical form and as a decimal (nearest hundredth).
  2. A pendulum must have a period of 1.5 seconds. Using \(T=2\pi\sqrt{L/9.8}\), find the required length to the nearest hundredth of a meter.
  3. An object falls from a building. Using \(t=\sqrt{h/16}\), find the height if the fall takes 5 seconds.

Answers may be placed in the answer key.

Bridge Note — Chapter 9
From Radical Functions to Exponential Functions

Chapter 8 developed your ability to work with expressions and functions that involve roots — which are, at their core, fractional powers. You have seen that the notation \(a^{m/n}\) connects roots and powers in a single unified language, and that this language obeys the same algebraic laws as integer exponents.

In Chapter 9, you will study exponential functions of the form \(f(x) = a \cdot b^x\), where the variable appears in the exponent rather than the base. These functions describe repeated multiplicative change: population growth, compound interest, radioactive decay, and many other real-world phenomena.

The connection runs deeper than it may first appear. The function \(f(x)=2^x\) is defined not only for integer values of x, but for all real numbers including fractions and irrationals. When \(x=1/2\), then \(2^{1/2}=\sqrt{2}\). When \(x=2/3\), then \(2^{2/3}=\sqrt[3]{4}\). The rational exponent work of this chapter is precisely what makes exponential functions continuous and defined at every real input.

In Chapter 10, logarithmic functions will appear as the inverses of exponential functions — just as radical functions are the inverses of power functions. The theme of inverse relationships that runs through Chapter 8 will continue throughout the rest of this textbook. Carry forward: rational exponents unify roots and powers; domains must be checked; inverse relationships organize entire families of functions; and verifying solutions is never optional.

MGU Library Connection
  • Radical and Rational Exponent Rules Reference — Appendix H of this textbook
  • Radical simplification and operations connect to MGU Algebra I, Chapter 13: Radicals and Rational Exponents
  • Radical models appear in MGU Physics I (pendulum, wave speed) and MGU Engineering Foundations
  • The distance formula reappears in MGU Trigonometry, Chapter 1, and MGU Precalculus, Chapter 3 (Conic Sections)
  • Standard deviation in MGU Statistics, Chapter 4, also involves a square root — connecting algebra to data analysis
  • Chapter 9 of this book: Exponential Functions — where rational exponents define continuity of exponential models
  • Chapter 10 of this book: Logarithmic Functions — where the inverse-function theme of Chapter 8 returns