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학습 가이드 > College Algebra

Radicals and Rational Exponents

A hardware store sells 16-ft ladders and 24-ft ladders. A window is located 12 feet above the ground. A ladder needs to be purchased that will reach the window from a point on the ground 5 feet from the building. To find out the length of ladder needed, we can draw a right triangle as shown in Figure 1, and use the Pythagorean Theorem.
A right triangle with a base of 5 feet, a height of 12 feet, and a hypotenuse labeled c Figure 1

a2+b2=c252+122=c2169=c2\begin{array}{ccc}\hfill {a}^{2}+{b}^{2}& =& {c}^{2}\hfill \\ \hfill {5}^{2}+{12}^{2}& =& {c}^{2}\hfill \\ \hfill 169& =& {c}^{2}\hfill \end{array}

Now, we need to find out the length that, when squared, is 169, to determine which ladder to choose. In other words, we need to find a square root. In this section, we will investigate methods of finding solutions to problems such as this one.

Evaluate and Simplify Square Roots

When the square root of a number is squared, the result is the original number. Since 42=16{4}^{2}=16, the square root of 1616 is 44. The square root function is the inverse of the squaring function just as subtraction is the inverse of addition. To undo squaring, we take the square root. In general terms, if aa is a positive real number, then the square root of aa is a number that, when multiplied by itself, gives aa. The square root could be positive or negative because multiplying two negative numbers gives a positive number. The principal square root is the nonnegative number that when multiplied by itself equals aa. The square root obtained using a calculator is the principal square root. The principal square root of aa is written as a\sqrt{a}. The symbol is called a radical, the term under the symbol is called the radicand, and the entire expression is called a radical expression. The expression: square root of twenty-five is enclosed in a circle. The circle has an arrow pointing to it labeled: Radical expression. The square root symbol has an arrow pointing to it labeled: Radical. The number twenty-five has an arrow pointing to it labeled: Radicand.

A General Note: Principal Square Root

The principal square root of aa is the nonnegative number that, when multiplied by itself, equals aa. It is written as a radical expression, with a symbol called a radical over the term called the radicand: a\sqrt{a}.

Q & A

Does 25=±5\sqrt{25}=\pm 5?

No. Although both 52{5}^{2} and (−5)2{\left(-5\right)}^{2} are 2525, the radical symbol implies only a nonnegative root, the principal square root. The principal square root of 25 is 25=5\sqrt{25}=5.

Example: Evaluating Square Roots

Evaluate each expression.
  1. 100\sqrt{100}
  2. 16\sqrt{\sqrt{16}}
  3. 25+144\sqrt{25+144}
  4. 49−81\sqrt{49}-\sqrt{81}

Answer:

  1. 100=10\sqrt{100}=10 because 102=100{10}^{2}=100
  2. 16=4=2\sqrt{\sqrt{16}}=\sqrt{4}=2 because 42=16{4}^{2}=16 and 22=4{2}^{2}=4
  3. 25+144=169=13\sqrt{25+144}=\sqrt{169}=13 because 132=169{13}^{2}=169
  4. 49−81=7−9=−2\sqrt{49}-\sqrt{81}=7 - 9=-2 because 72=49{7}^{2}=49 and 92=81{9}^{2}=81

Q & A

For 25+144\sqrt{25+144}, can we find the square roots before adding?

No. 25+144=5+12=17\sqrt{25}+\sqrt{144}=5+12=17. This is not equivalent to 25+144=13\sqrt{25+144}=13. The order of operations requires us to add the terms in the radicand before finding the square root.

Try It

Evaluate each expression.
  1. 225\sqrt{225}
  2. 81\sqrt{\sqrt{81}}
  3. 25−9\sqrt{25 - 9}
  4. 36+121\sqrt{36}+\sqrt{121}

Answer:

  1. 1515
  2. 33
  3. 44
  4. 1717

Use the Product Rule to Simplify Square Roots

To simplify a square root, we rewrite it such that there are no perfect squares in the radicand. There are several properties of square roots that allow us to simplify complicated radical expressions. The first rule we will look at is the product rule for simplifying square roots, which allows us to separate the square root of a product of two numbers into the product of two separate rational expressions. For instance, we can rewrite 15\sqrt{15} as 3⋅5\sqrt{3}\cdot \sqrt{5}. We can also use the product rule to express the product of multiple radical expressions as a single radical expression.

A General Note: The Product Rule for Simplifying Square Roots

If aa and bb are nonnegative, the square root of the product abab is equal to the product of the square roots of aa and bb.
ab=a⋅b\sqrt{ab}=\sqrt{a}\cdot \sqrt{b}

How To: Given a square root radical expression, use the product rule to simplify it.

  1. Factor any perfect squares from the radicand.
  2. Write the radical expression as a product of radical expressions.
  3. Simplify.

Example: Using the Product Rule to Simplify Square Roots

Simplify the radical expression.
  1. 300\sqrt{300}
  2. 162a5b4\sqrt{162{a}^{5}{b}^{4}}

Answer:

  1. 100⋅3Factor perfect square from radicand.100⋅3Write radical expression as product of radical expressions.103Simplify. \begin{array}{cc}\sqrt{100\cdot 3}\hfill & \text{Factor perfect square from radicand}.\hfill \\ \sqrt{100}\cdot \sqrt{3}\hfill & \text{Write radical expression as product of radical expressions}.\hfill \\ 10\sqrt{3}\hfill & \text{Simplify}.\hfill \\ \text{ }\end{array}
  2. 81a4b4⋅2aFactor perfect square from radicand.81a4b4⋅2aWrite radical expression as product of radical expressions.9a2b22aSimplify.\begin{array}{cc}\sqrt{81{a}^{4}{b}^{4}\cdot 2a}\hfill & \text{Factor perfect square from radicand}.\hfill \\ \sqrt{81{a}^{4}{b}^{4}}\cdot \sqrt{2a}\hfill & \text{Write radical expression as product of radical expressions}.\hfill \\ 9{a}^{2}{b}^{2}\sqrt{2a}\hfill & \text{Simplify}.\hfill \end{array}

Try It

Simplify 50x2y3z\sqrt{50{x}^{2}{y}^{3}z}.

Answer: 5∣x∣∣y∣2yz5|x||y|\sqrt{2yz}. Notice the absolute value signs around x and y? That’s because their value must be positive!

Using the Quotient Rule to Simplify Square Roots

Just as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the quotient rule for simplifying square roots. It can be helpful to separate the numerator and denominator of a fraction under a radical so that we can take their square roots separately. We can rewrite 52\sqrt{\frac{5}{2}} as 52\frac{\sqrt{5}}{\sqrt{2}}.

A General Note: The Quotient Rule for Simplifying Square Roots

The square root of the quotient ab\frac{a}{b} is equal to the quotient of the square roots of aa and bb, where b≠0b\ne 0.
ab=ab\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}

How To: Given a radical expression, use the quotient rule to simplify it.

  1. Write the radical expression as the quotient of two radical expressions.
  2. Simplify the numerator and denominator.

Example: Using the Quotient Rule to Simplify Square Roots

Simplify the radical expression.

536\sqrt{\frac{5}{36}}

Answer:

536Write as quotient of two radical expressions.56Simplify denominator.\begin{array}{cc}\frac{\sqrt{5}}{\sqrt{36}}\hfill & \text{Write as quotient of two radical expressions}.\hfill \\ \frac{\sqrt{5}}{6}\hfill & \text{Simplify denominator}.\hfill \end{array}

Try It

Simplify 2x29y4\sqrt{\frac{2{x}^{2}}{9{y}^{4}}}.

Answer: x23y2\frac{x\sqrt{2}}{3{y}^{2}}. We do not need the absolute value signs for y2{y}^{2} because that term will always be nonnegative.

Operations on Square Roots

We can add or subtract radical expressions only when they have the same radicand and when they have the same radical type such as square roots. For example, the sum of 2\sqrt{2} and 323\sqrt{2} is 424\sqrt{2}. However, it is often possible to simplify radical expressions, and that may change the radicand. The radical expression 18\sqrt{18} can be written with a 22 in the radicand, as 323\sqrt{2}, so 2+18=2+32=42\sqrt{2}+\sqrt{18}=\sqrt{2}+3\sqrt{2}=4\sqrt{2}.

How To: Given a radical expression requiring addition or subtraction of square roots, solve.

  1. Simplify each radical expression.
  2. Add or subtract expressions with equal radicands.

Example: Adding Square Roots

Add 512+235\sqrt{12}+2\sqrt{3}.

Answer: We can rewrite 5125\sqrt{12} as 54⋅35\sqrt{4\cdot 3}. According the product rule, this becomes 5435\sqrt{4}\sqrt{3}. The square root of 4\sqrt{4} is 2, so the expression becomes 5(2)35\left(2\right)\sqrt{3}, which is 10310\sqrt{3}. Now we can the terms have the same radicand so we can add.

103+23=12310\sqrt{3}+2\sqrt{3}=12\sqrt{3}

Try It

Add 5+620\sqrt{5}+6\sqrt{20}.

Answer: 13513\sqrt{5}

Rationalize Denominators

When an expression involving square root radicals is written in simplest form, it will not contain a radical in the denominator. We can remove radicals from the denominators of fractions using a process called rationalizing the denominator. We know that multiplying by 1 does not change the value of an expression. We use this property of multiplication to change expressions that contain radicals in the denominator. To remove radicals from the denominators of fractions, multiply by the form of 1 that will eliminate the radical. For a denominator containing a single term, multiply by the radical in the denominator over itself. In other words, if the denominator is bcb\sqrt{c}, multiply by cc\frac{\sqrt{c}}{\sqrt{c}}. For a denominator containing the sum or difference of a rational and an irrational term, multiply the numerator and denominator by the conjugate of the denominator, which is found by changing the sign of the radical portion of the denominator. If the denominator is a+bca+b\sqrt{c}, then the conjugate is a−bca-b\sqrt{c}.

How To: Given an expression with a single square root radical term in the denominator, rationalize the denominator.

  1. Multiply the numerator and denominator by the radical in the denominator.
  2. Simplify.

Example: Rationalizing a Denominator Containing a Single Term

Write 23310\frac{2\sqrt{3}}{3\sqrt{10}} in simplest form.

Answer: The radical in the denominator is 10\sqrt{10}. So multiply the fraction by 1010\frac{\sqrt{10}}{\sqrt{10}}. Then simplify.

23310⋅1010 23030 3015\begin{array}{l}\frac{2\sqrt{3}}{3\sqrt{10}}\cdot \frac{\sqrt{10}}{\sqrt{10}}\text{ }\\ \frac{2\sqrt{30}}{30}\text{ }\\ \frac{\sqrt{30}}{15}\end{array}

Try It

Write 1232\frac{12\sqrt{3}}{\sqrt{2}} in simplest form.

Answer: 666\sqrt{6}

Nth Roots and Rational Exponents

Using Rational Roots

Although square roots are the most common rational roots, we can also find cube roots, 4th roots, 5th roots, and more. Just as the square root function is the inverse of the squaring function, these roots are the inverse of their respective power functions. These functions can be useful when we need to determine the number that, when raised to a certain power, gives a certain number. Suppose we know that a3=8{a}^{3}=8. We want to find what number raised to the 3rd power is equal to 8. Since 23=8{2}^{3}=8, we say that 2 is the cube root of 8. The nth root of aa is a number that, when raised to the nth power, gives aa. For example, −3-3 is the 5th root of −243-243 because (−3)5=−243{\left(-3\right)}^{5}=-243. If aa is a real number with at least one nth root, then the principal nth root of aa is the number with the same sign as aa that, when raised to the nth power, equals aa. The principal nth root of aa is written as an\sqrt[n]{a}, where nn is a positive integer greater than or equal to 2. In the radical expression, nn is called the index of the radical.

A General Note: Principal nth Root

If aa is a real number with at least one nth root, then the principal nth root of aa, written as an\sqrt[n]{a}, is the number with the same sign as aa that, when raised to the nth power, equals aa. The index of the radical is nn.

Example: Simplifying nth Roots

Simplify each of the following:
  1. −325\sqrt[5]{-32}
  2. 44⋅1,0244\sqrt[4]{4}\cdot \sqrt[4]{1,024}
  3. −8x61253-\sqrt[3]{\frac{8{x}^{6}}{125}}
  4. 834−4848\sqrt[4]{3}-\sqrt[4]{48}

Answer:

  1. −325=−2\sqrt[5]{-32}=-2 because (−2)5=−32 {\left(-2\right)}^{5}=-32 \\ \text{ }
  2. First, express the product as a single radical expression. 4,0964=8\sqrt[4]{4,096}=8 because 84=4,096{8}^{4}=4,096
  3. −8x631253Write as quotient of two radical expressions.−2x25Simplify.\begin{array}{cc}\\ \frac{-\sqrt[3]{8{x}^{6}}}{\sqrt[3]{125}}\hfill & \text{Write as quotient of two radical expressions}.\hfill \\ \frac{-2{x}^{2}}{5}\hfill & \text{Simplify}.\hfill \\ \end{array}
  4. 834−234Simplify to get equal radicands.634Add.\begin{array}{cc}\\ 8\sqrt[4]{3}-2\sqrt[4]{3}\hfill & \text{Simplify to get equal radicands}.\hfill \\ 6\sqrt[4]{3} \hfill & \text{Add}.\hfill \\ \end{array}

Try It

Simplify.
  1. −2163\sqrt[3]{-216}
  2. 380454\frac{3\sqrt[4]{80}}{\sqrt[4]{5}}
  3. 69,0003+757636\sqrt[3]{9,000}+7\sqrt[3]{576}

Answer:

  1. −6-6
  2. 66
  3. 889388\sqrt[3]{9}

Using Rational Exponents

Radical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the index nn is even, then aa cannot be negative.
a1n=an{a}^{\frac{1}{n}}=\sqrt[n]{a}
We can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in lowest terms. We raise the base to a power and take an nth root. The numerator tells us the power and the denominator tells us the root.
amn=(an)m=amn{a}^{\frac{m}{n}}={\left(\sqrt[n]{a}\right)}^{m}=\sqrt[n]{{a}^{m}}
All of the properties of exponents that we learned for integer exponents also hold for rational exponents.

Example: Rational Exponents

Rational exponents are another way to express principal nth roots. The general form for converting between a radical expression with a radical symbol and one with a rational exponent is
amn=(an)m=amn{a}^{\frac{m}{n}}={\left(\sqrt[n]{a}\right)}^{m}=\sqrt[n]{{a}^{m}}

How To: Given an expression with a rational exponent, write the expression as a radical.

  1. Determine the power by looking at the numerator of the exponent.
  2. Determine the root by looking at the denominator of the exponent.
  3. Using the base as the radicand, raise the radicand to the power and use the root as the index.

Example: Writing Rational Exponents as Radicals

Write 34323{343}^{\frac{2}{3}} as a radical. Simplify.

Answer: The 2 tells us the power and the 3 tells us the root.

34323=(3433)2=34323{343}^{\frac{2}{3}}={\left(\sqrt[3]{343}\right)}^{2}=\sqrt[3]{{343}^{2}}

We know that 3433=7\sqrt[3]{343}=7 because 73=343{7}^{3}=343. Because the cube root is easy to find, it is easiest to find the cube root before squaring for this problem. In general, it is easier to find the root first and then raise it to a power.

34323=(3433)2=72=49{343}^{\frac{2}{3}}={\left(\sqrt[3]{343}\right)}^{2}={7}^{2}=49

Try It

Write 952{9}^{\frac{5}{2}} as a radical. Simplify.

Answer: (9)5=35=243{\left(\sqrt{9}\right)}^{5}={3}^{5}=243

Key Concepts

  • The principal square root of a number aa is the nonnegative number that when multiplied by itself equals aa.
  • If aa and bb are nonnegative, the square root of the product abab is equal to the product of the square roots of aa and bb
  • If aa and bb are nonnegative, the square root of the quotient ab\frac{a}{b} is equal to the quotient of the square roots of aa and bb
  • We can add and subtract radical expressions if they have the same radicand and the same index.
  • Radical expressions written in simplest form do not contain a radical in the denominator. To eliminate the square root radical from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator.
  • The principal nth root of aa is the number with the same sign as aa that when raised to the nth power equals aa. These roots have the same properties as square roots.
  • Radicals can be rewritten as rational exponents and rational exponents can be rewritten as radicals.
  • The properties of exponents apply to rational exponents.

Glossary

index the number above the radical sign indicating the nth root principal nth root the number with the same sign as aa that when raised to the nth power equals aa principal square root the nonnegative square root of a number aa that, when multiplied by itself, equals aa radical the symbol used to indicate a root radical expression an expression containing a radical symbol radicand the number under the radical symbol

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