The Central Idea
A radical expression is simplified by separating perfect powers from the part of the radicand that cannot be simplified further.
For example,
\[ \sqrt{72} \]can be rewritten as
\[ \sqrt{36\cdot2}. \]Since
\[ \sqrt{36}=6, \]we obtain
\[ \boxed{\sqrt{72}=6\sqrt2}. \]The exact value has not changed. Only its form has.
What Does It Mean to Simplify a Radical?
A radical expression is usually considered simplified when:
- the radicand contains no removable perfect-power factor;
- fractions under the radical have been handled appropriately;
- like radicals have been combined;
- coefficients have been simplified;
- the expression remains exact;
- domain restrictions are preserved.
Later, denominators containing radicals will also be rationalized.
Perfect Squares Inside Square Roots
To simplify a square root, look for a perfect-square factor.
For example,
\[ \sqrt{48}. \]Since
\[ 48=16\cdot3, \]we have
\[ \sqrt{48} = \sqrt{16\cdot3} = \sqrt{16}\sqrt3. \]Therefore,
\[ \boxed{\sqrt{48}=4\sqrt3}. \]Use the Largest Convenient Perfect Square
Consider
\[ \sqrt{180}. \]A large perfect-square factor is
\[ 36. \]Therefore,
\[ \sqrt{180} = \sqrt{36\cdot5} = 6\sqrt5. \]Thus,
\[ \boxed{\sqrt{180}=6\sqrt5}. \]Prime Factorization Method
Prime factorization provides a systematic method.
Consider
\[ \sqrt{72}. \]Factor:
\[ 72=2^3\cdot3^2. \]Pair equal prime factors:
\[ \sqrt{2^2\cdot2\cdot3^2}. \]Each complete pair contributes one factor outside:
\[ \boxed{6\sqrt2}. \]Why Pairs Matter for Square Roots
A square root reverses squaring.
Therefore two copies of a factor under a square root become one copy outside:
\[ \sqrt{a^2}=|a| \]for real \(a\).
With known nonnegative quantities, this often appears simply as
\[ \sqrt{a^2}=a. \]The absolute-value distinction remains important when variables may be negative.
Simplifying Cube Roots
For cube roots, look for groups of three equal factors.
Consider
\[ \sqrt[3]{54}. \]Since
\[ 54=27\cdot2, \]we have
\[ \sqrt[3]{54} = \sqrt[3]{27}\sqrt[3]{2} = 3\sqrt[3]{2}. \]Therefore,
\[ \boxed{\sqrt[3]{54}=3\sqrt[3]{2}}. \]Simplifying Higher Roots
The same principle works for any index.
For an \(n\)th root, groups of \(n\) equal factors can be extracted.
For example,
\[ \sqrt[4]{48} = \sqrt[4]{16\cdot3}. \]Since
\[ 16=2^4, \]we obtain
\[ \boxed{\sqrt[4]{48}=2\sqrt[4]{3}}. \]The Product Rule for Radicals
For nonnegative real \(a\) and \(b\),
\[ \boxed{ \sqrt{ab}=\sqrt a\,\sqrt b }. \]More generally, for compatible real-domain conditions,
\[ \boxed{ \sqrt[n]{ab} = \sqrt[n]{a}\sqrt[n]{b} }. \]This rule allows perfect-power factors to be separated from a radicand.
Using the Product Rule in Reverse
The same identity can combine radicals:
\[ \sqrt a\,\sqrt b = \sqrt{ab}. \]For example,
\[ \sqrt2\sqrt6 = \sqrt{12} = 2\sqrt3. \]Therefore,
\[ \boxed{\sqrt2\sqrt6=2\sqrt3}. \]The Quotient Rule for Radicals
For suitable real values with \(b>0\),
\[ \boxed{ \sqrt{\frac ab} = \frac{\sqrt a}{\sqrt b} }. \]More generally,
\[ \sqrt[n]{\frac ab} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \]whenever the radicals are real and the denominator is nonzero.
Example: Simplify a Radical Fraction
Simplify
\[ \sqrt{\frac{25}{49}}. \]Apply the quotient rule:
\[ \frac{\sqrt{25}}{\sqrt{49}} = \frac57. \]Therefore,
\[ \boxed{\sqrt{\frac{25}{49}}=\frac57}. \]Coefficients Outside Radicals
A number multiplying a radical is a coefficient.
For example, in
\[ 5\sqrt3, \]the coefficient is \(5\), and the radical part is
\[ \sqrt3. \]Keeping these roles separate makes addition and subtraction easier to understand.
Like Radicals
Like radicals have the same index and the same simplified radicand.
Examples include
\[ 3\sqrt5 \]and
\[ 7\sqrt5. \]Since their radical parts are identical, their coefficients can be combined.
Adding Like Radicals
Consider
\[ 3\sqrt5+7\sqrt5. \]Factor out the common radical:
\[ (3+7)\sqrt5. \]Therefore,
\[ \boxed{3\sqrt5+7\sqrt5=10\sqrt5}. \]Subtracting Like Radicals
Consider
\[ 9\sqrt2-4\sqrt2. \]Combine the coefficients:
\[ (9-4)\sqrt2. \]Therefore,
\[ \boxed{9\sqrt2-4\sqrt2=5\sqrt2}. \]Unlike Radicals Cannot Be Combined Directly
Consider
\[ 3\sqrt2+5\sqrt3. \]The radical parts are different.
Therefore the terms are not like radicals and cannot be combined by adding coefficients.
The expression is already simplified with respect to addition:
\[ \boxed{3\sqrt2+5\sqrt3}. \]Simplify Before Deciding Whether Radicals Are Like
Consider
\[ \sqrt8+\sqrt{18}. \]At first the radicands differ.
Simplify:
\[ \sqrt8 = 2\sqrt2, \] \[ \sqrt{18} = 3\sqrt2. \]Now the radicals are like:
\[ 2\sqrt2+3\sqrt2=5\sqrt2. \]Therefore,
\[ \boxed{\sqrt8+\sqrt{18}=5\sqrt2}. \]Example: Subtract After Simplifying
Simplify
\[ \sqrt{50}-\sqrt8. \]Rewrite:
\[ \sqrt{50}=5\sqrt2, \] \[ \sqrt8=2\sqrt2. \]Therefore,
\[ 5\sqrt2-2\sqrt2=3\sqrt2. \]Thus,
\[ \boxed{\sqrt{50}-\sqrt8=3\sqrt2}. \]Multiplying a Number by a Radical
Ordinary multiplication applies to coefficients.
For example,
\[ 4(3\sqrt5) = 12\sqrt5. \]Therefore,
\[ \boxed{4(3\sqrt5)=12\sqrt5}. \]Multiplying Two Radical Terms
Consider
\[ (2\sqrt3)(5\sqrt6). \]Multiply the coefficients:
\[ 2\cdot5=10. \]Multiply the radicals:
\[ \sqrt3\sqrt6=\sqrt{18}. \]Simplify:
\[ \sqrt{18}=3\sqrt2. \]Therefore,
\[ \boxed{(2\sqrt3)(5\sqrt6)=30\sqrt2}. \]Squaring a Radical
Because squaring and taking a principal square root undo each other for a nonnegative radicand,
\[ \left(\sqrt a\right)^2=a \]for \(a\geq0\).
For example,
\[ (\sqrt7)^2=7. \]Likewise,
\[ (3\sqrt5)^2 = 9(5) = 45. \]Multiplying Binomials Containing Radicals
Distributive multiplication works normally.
Consider
\[ (2+\sqrt3)(4+\sqrt5). \]Distribute:
\[ 8+2\sqrt5+4\sqrt3+\sqrt{15}. \]No radical terms are like, so the result is
\[ \boxed{ 8+2\sqrt5+4\sqrt3+\sqrt{15} }. \]Example: Binomial Multiplication with Like Radicals
Multiply
\[ (3+\sqrt2)(2+\sqrt2). \]Distribute:
\[ 6+3\sqrt2+2\sqrt2+2. \]Combine like terms:
\[ 8+5\sqrt2. \]Therefore,
\[ \boxed{(3+\sqrt2)(2+\sqrt2)=8+5\sqrt2}. \]Dividing Radical Terms
Consider
\[ \frac{6\sqrt{15}}{3\sqrt5}. \]Reduce the coefficients:
\[ \frac63=2. \]Divide the radicals:
\[ \frac{\sqrt{15}}{\sqrt5} = \sqrt3. \]Therefore,
\[ \boxed{ \frac{6\sqrt{15}}{3\sqrt5}=2\sqrt3 }. \]Simplifying Variables Under Square Roots
Variables can contain perfect-square factors just as numbers do.
For example,
\[ \sqrt{x^2} = |x|. \]If a problem states or assumes
\[ x\geq0, \]then
\[ \sqrt{x^2}=x. \]Without that assumption, the absolute value must be retained.
Example: Variable Powers Under a Square Root
Simplify
\[ \sqrt{x^6} \]for real \(x\).
Write
\[ x^6=(x^3)^2. \]Then
\[ \sqrt{x^6} = |x^3|. \]Therefore,
\[ \boxed{\sqrt{x^6}=|x^3|}. \]Example: Numerical and Variable Factors
Simplify
\[ \sqrt{72x^2} \]for real \(x\).
Factor:
\[ 72x^2=36\cdot2\cdot x^2. \]Therefore,
\[ \sqrt{72x^2} = 6|x|\sqrt2. \]Thus,
\[ \boxed{\sqrt{72x^2}=6|x|\sqrt2}. \]Odd Roots of Variable Powers
Odd roots preserve sign, so absolute value is not required in the same way.
For example,
\[ \sqrt[3]{x^3}=x \]for every real \(x\).
Likewise,
\[ \sqrt[3]{x^6} = x^2. \]Example: Simplify a Cube Root with Variables
Simplify
\[ \sqrt[3]{54x^4}. \]Factor:
\[ 54x^4 = 27x^3(2x). \]Extract the perfect cubes:
\[ \sqrt[3]{27x^3}\sqrt[3]{2x} = 3x\sqrt[3]{2x}. \]Therefore,
\[ \boxed{ \sqrt[3]{54x^4} = 3x\sqrt[3]{2x} }. \]Exponent Notation Can Help Simplification
Rational exponents provide another way to see why factors come out of radicals.
For example,
\[ \sqrt{x^5} = x^{5/2} \]on the appropriate real domain.
Since
\[ \frac52=2+\frac12, \]we can write
\[ x^{5/2} = x^2x^{1/2} = x^2\sqrt x. \]Simplifying \(\sqrt{x^5}\)
Factor directly:
\[ x^5=x^4x. \]Then
\[ \sqrt{x^5} = \sqrt{x^4}\sqrt x. \]Since
\[ \sqrt{x^4}=x^2, \]we obtain
\[ \boxed{ \sqrt{x^5}=x^2\sqrt x } \]on the real domain \(x\geq0\).
Simplifying a Radical with Several Variables
Suppose \(x\geq0\) and \(y\geq0\). Simplify
\[ \sqrt{48x^5y^3}. \]Factor:
\[ 48x^5y^3 = 16x^4y^2(3xy). \]Then
\[ \sqrt{48x^5y^3} = 4x^2y\sqrt{3xy}. \]Therefore,
\[ \boxed{ \sqrt{48x^5y^3} = 4x^2y\sqrt{3xy} }. \]Adding Variable Radicals
Suppose \(x\geq0\). Simplify
\[ 3\sqrt{2x}+5\sqrt{2x}. \]The radical parts match exactly.
Therefore,
\[ (3+5)\sqrt{2x} = 8\sqrt{2x}. \]Thus,
\[ \boxed{8\sqrt{2x}}. \]Simplify Before Combining Variable Radicals
Suppose \(x\geq0\). Simplify
\[ \sqrt{8x}+\sqrt{18x}. \]Rewrite:
\[ \sqrt{8x} = 2\sqrt{2x}, \] \[ \sqrt{18x} = 3\sqrt{2x}. \]Therefore,
\[ \boxed{ \sqrt{8x}+\sqrt{18x} = 5\sqrt{2x} }. \]Radical Expressions and Exact Arithmetic
Radical forms often preserve exact information better than decimal approximations.
For example,
\[ 3\sqrt2 \]is exact.
The approximation
\[ 3\sqrt2\approx4.243 \]is useful numerically but no longer exact.
Algebraic simplification generally keeps radicals exact until an approximation is specifically needed.
Why \(\sqrt{a+b}\) Does Not Split
The product rule applies to multiplication:
\[ \sqrt{ab} = \sqrt a\sqrt b. \]It does not apply to addition:
\[ \boxed{ \sqrt{a+b}\neq\sqrt a+\sqrt b } \]in general.
For example,
\[ \sqrt{9+16}=5, \]while
\[ \sqrt9+\sqrt{16}=7. \]Why \(\sqrt{a-b}\) Does Not Split
Likewise,
\[ \boxed{ \sqrt{a-b}\neq\sqrt a-\sqrt b } \]in general.
For example,
\[ \sqrt{25-9}=4, \]while
\[ \sqrt{25}-\sqrt9=2. \]Do Not Add Radicands When Adding Radicals
It is incorrect to write
\[ \sqrt2+\sqrt3=\sqrt5. \]Addition of radical expressions does not combine radicands this way.
Since \(\sqrt2\) and \(\sqrt3\) are unlike radicals, the exact simplified form remains
\[ \boxed{\sqrt2+\sqrt3}. \]Multiplication Is Different from Addition
Although
\[ \sqrt2+\sqrt3 \]cannot be merged into one radical, the product
\[ \sqrt2\sqrt3 \]can be combined:
\[ \sqrt2\sqrt3=\sqrt6. \]Radical multiplication and radical addition follow different rules.
Common Error: Extracting a Factor That Is Not a Perfect Power
Consider
\[ \sqrt{12}. \]We may factor
\[ 12=4\cdot3 \]and extract \(\sqrt4=2\).
But we cannot simply pull out \(3\) because \(3^2\) is not a factor of \(12\).
The correct result is
\[ \boxed{2\sqrt3}. \]Common Error: Combining Unlike Radicals
The expression
\[ 2\sqrt3+5\sqrt2 \]does not equal
\[ 7\sqrt5. \]The radical parts differ, so the terms are not like radicals.
The expression cannot be combined by addition.
Common Error: Forgetting to Simplify First
Consider
\[ \sqrt{12}+\sqrt{27}. \]The terms initially look unlike.
Simplify:
\[ \sqrt{12}=2\sqrt3, \] \[ \sqrt{27}=3\sqrt3. \]Then
\[ 2\sqrt3+3\sqrt3=5\sqrt3. \]Therefore,
\[ \boxed{5\sqrt3}. \]Common Error: Writing \(\sqrt{x^2}=x\) Without Conditions
For real \(x\),
\[ \boxed{\sqrt{x^2}=|x|}. \]If the problem states \(x\geq0\), then \(|x|=x\).
Without that information, dropping the absolute value can produce an incorrect result.
Common Error: Applying the Product Rule Outside Its Real Domain
Over the real numbers, square-root identities require appropriate nonnegative radicands.
For example,
\[ \sqrt{(-1)(-1)} = \sqrt1 = 1, \]while neither
\[ \sqrt{-1} \]is real.
Therefore the real square-root product rule should not be applied blindly to negative radicands.
Common Error: Converting to Decimals Too Early
Suppose an exact calculation contains
\[ \sqrt8+\sqrt{18}. \]Converting both values to decimals and adding introduces approximation.
Simplifying exactly gives
\[ 2\sqrt2+3\sqrt2 = 5\sqrt2. \]Approximate only when the problem requires a numerical estimate.
A Reliable Procedure for Simplifying Square Roots
- Factor the radicand.
- Identify the largest convenient perfect-square factor.
- Use the product rule to separate that factor.
- Evaluate the perfect square root.
- Repeat if another perfect-square factor remains.
- For variable squares, use absolute value unless nonnegativity is known.
A Reliable Procedure for Higher Roots
- Identify the index \(n\).
- Factor the radicand.
- Look for perfect \(n\)th-power factors.
- Extract one factor for every complete group of \(n\) equal factors.
- Preserve any remaining factors inside the radical.
- Check sign and domain conditions appropriate to the index.
A Reliable Procedure for Adding and Subtracting Radicals
- Simplify every radical separately.
- Identify terms with the same index and simplified radicand.
- Combine only the coefficients of like radicals.
- Leave unlike radicals as separate terms.
- Check whether the resulting radicals can be simplified further.
A Reliable Procedure for Multiplying Radicals
- Multiply numerical coefficients.
- Multiply compatible radical parts using the product rule.
- Simplify the resulting radicand.
- For binomials, distribute each term carefully.
- Combine like radical terms after multiplication.
- Preserve domain restrictions.
A Reliable Procedure for Dividing Radicals
- Reduce numerical coefficients.
- Use the quotient rule when its domain conditions are satisfied.
- Simplify the resulting radical.
- Keep the denominator nonzero.
- If a radical remains in the denominator, prepare to rationalize it.
Example: Full Simplification Workflow
Simplify
\[ \sqrt{288}. \]Factor out a large perfect square:
\[ 288=144\cdot2. \]Therefore,
\[ \sqrt{288} = \sqrt{144}\sqrt2. \] \[ = 12\sqrt2. \]Thus,
\[ \boxed{\sqrt{288}=12\sqrt2}. \]Example: Full Addition Workflow
Simplify
\[ 2\sqrt{12}+3\sqrt{27}-\sqrt{75}. \]Simplify each radical:
\[ \sqrt{12}=2\sqrt3, \] \[ \sqrt{27}=3\sqrt3, \] \[ \sqrt{75}=5\sqrt3. \]Substitute:
\[ 2(2\sqrt3)+3(3\sqrt3)-5\sqrt3. \]Combine:
\[ 4\sqrt3+9\sqrt3-5\sqrt3 = 8\sqrt3. \]Therefore,
\[ \boxed{8\sqrt3}. \]Example: Full Multiplication Workflow
Simplify
\[ (3\sqrt2)(4\sqrt6). \]Multiply coefficients:
\[ 3\cdot4=12. \]Multiply radicals:
\[ \sqrt2\sqrt6=\sqrt{12}. \]Simplify:
\[ \sqrt{12}=2\sqrt3. \]Therefore,
\[ 12(2\sqrt3)=24\sqrt3. \]Thus,
\[ \boxed{24\sqrt3}. \]Example: Full Variable Workflow
Suppose \(x\geq0\). Simplify
\[ \sqrt{200x^5}. \]Factor:
\[ 200x^5 = 100x^4(2x). \]Therefore,
\[ \sqrt{200x^5} = 10x^2\sqrt{2x}. \]Thus,
\[ \boxed{ \sqrt{200x^5} = 10x^2\sqrt{2x} }. \]Practice 1: Simplify a Square Root
Simplify:
\[ \sqrt{75}. \]Practice 2: Simplify a Cube Root
Simplify:
\[ \sqrt[3]{128}. \]Practice 3: Add Like Radicals
Simplify:
\[ 4\sqrt7+9\sqrt7. \]Practice 4: Simplify Before Adding
Simplify:
\[ \sqrt{20}+\sqrt{45}. \]Practice 5: Subtract Radicals
Simplify:
\[ 3\sqrt{32}-2\sqrt8. \]Practice 6: Multiply Radical Terms
Simplify:
\[ (2\sqrt3)(5\sqrt{12}). \]Practice 7: Divide Radical Terms
Simplify:
\[ \frac{8\sqrt{18}}{4\sqrt2}. \]Practice 8: Variable Radical
Simplify for real \(x\):
\[ \sqrt{36x^2}. \]Practice 9: Cube Root with Variables
Simplify:
\[ \sqrt[3]{16x^4}. \]Practice 10: Identify the Error
A student writes
\[ \sqrt8+\sqrt2=\sqrt{10}. \]Explain the error and simplify correctly.
Practice Answers
Practice 1
Factor:
\[ 75=25\cdot3. \]Therefore,
\[ \sqrt{75} = 5\sqrt3. \]Thus,
\[ \boxed{5\sqrt3}. \]Practice 2
Factor:
\[ 128=64\cdot2. \]Since
\[ \sqrt[3]{64}=4, \]we obtain
\[ \boxed{4\sqrt[3]{2}}. \]Practice 3
The radicals are like:
\[ (4+9)\sqrt7=13\sqrt7. \]Therefore,
\[ \boxed{13\sqrt7}. \]Practice 4
Simplify:
\[ \sqrt{20}=2\sqrt5, \] \[ \sqrt{45}=3\sqrt5. \]Therefore,
\[ 2\sqrt5+3\sqrt5=5\sqrt5. \]Thus,
\[ \boxed{5\sqrt5}. \]Practice 5
Simplify:
\[ \sqrt{32}=4\sqrt2, \] \[ \sqrt8=2\sqrt2. \]Then
\[ 3(4\sqrt2)-2(2\sqrt2) = 12\sqrt2-4\sqrt2. \]Therefore,
\[ \boxed{8\sqrt2}. \]Practice 6
Multiply:
\[ (2\sqrt3)(5\sqrt{12}) = 10\sqrt{36}. \] \[ = 10(6). \]Therefore,
\[ \boxed{60}. \]Practice 7
Reduce coefficients:
\[ \frac84=2. \]Divide radicals:
\[ \frac{\sqrt{18}}{\sqrt2} = \sqrt9 = 3. \]Therefore,
\[ \boxed{6}. \]Practice 8
For real \(x\),
\[ \sqrt{36x^2} = 6|x|. \]Therefore,
\[ \boxed{6|x|}. \]Practice 9
Factor:
\[ 16x^4 = 8x^3(2x). \]Then
\[ \sqrt[3]{16x^4} = 2x\sqrt[3]{2x}. \]Therefore,
\[ \boxed{2x\sqrt[3]{2x}}. \]Practice 10
Radicands cannot be added merely because two square roots are being added.
Simplify \(\sqrt8\):
\[ \sqrt8=2\sqrt2. \]Then
\[ 2\sqrt2+\sqrt2=3\sqrt2. \]Therefore,
\[ \boxed{3\sqrt2}. \]What to Remember
Radical simplification is based on extracting perfect powers and using radical laws carefully.
The most useful identities include
\[ \boxed{ \sqrt{ab} = \sqrt a\,\sqrt b } \]under the appropriate real-domain conditions, and
\[ \boxed{ \sqrt{\frac ab} = \frac{\sqrt a}{\sqrt b} }. \]The essential habits are:
- factor radicands to expose perfect powers;
- extract one factor for each complete group determined by the index;
- simplify radicals before deciding whether they are like;
- combine only radicals with the same index and simplified radicand;
- multiply compatible radicals using the product rule;
- divide compatible radicals using the quotient rule;
- use absolute value when simplifying even roots of variable even powers unless nonnegativity is known;
- keep exact radical form until an approximation is actually needed;
- do not distribute radicalization across addition or subtraction.
The central habit is: simplify the structure inside each radical first, then perform arithmetic only on expressions whose forms are algebraically compatible.
Where We Go Next
Radical expressions can now be simplified, added, subtracted, multiplied, and divided. One important issue remains: traditional simplified form usually avoids radicals in denominators.
The next lesson, Conjugates and Rationalizing Denominators, develops the conjugate relationship and shows how multiplication by carefully chosen forms of \(1\) can remove radicals from denominators while preserving the value of an expression.