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1.3: Radicals and Rational Exponents

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Learning Objectives
  • Evaluate square roots.
  • Use the product rule to simplify square roots.
  • Use the quotient rule to simplify square roots.
  • Add and subtract square roots.
  • Rationalize denominators.
  • Use rational roots.

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 1.3.1, 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 1.3.1: A right triangle

a2+b2=c252+122=c2169=c2

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.

Evaluating Square Roots

When the square root of a number is squared, the result is the original number. Since 42=16, the square root of 16 is 4.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 a is a positive real number, then the square root of a is a number that, when multiplied by itself, gives a.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 a. The square root obtained using a calculator is the principal square root.

The principal square root of a is written as 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.

Example 1.3.1

Does 25=±5?

Solution

No. Although both 52 and (5)2 are 25, the radical symbol implies only a nonnegative root, the principal square root. The principal square root of 25 is 25=5.

Note

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

Example 1.3.2: Evaluating Square Roots

Evaluate each expression.

  1. 16
  2. 49-81

Solution

  1. 16=4=2 because 42=16 and 22=4
  2. 4981=79=2 because 72=49 and 92=81
Example 1.3.3:

For 25+144,can we find the square roots before adding?

Solution

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

Exercise 1.3.1

Evaluate each expression.

  1. 81
  2. 36+121
 
Answer a

3

Answer b

17

Using 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 as 3×5. We can also use the product rule to express the product of multiple radical expressions as a single radical expression.

The Product Rule For Simplifying Square Roots

If a and b are nonnegative, the square root of the product ab is equal to the product of the square roots of a and b

ab=a×b

HOWTO: 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 1.3.4: Using the Product Rule to Simplify Square Roots

Simplify the radical expression.

  1. 300
  2. 162a5b4
Solution

a. 100×3 Factor perfect square from radicand.

100×3 Write radical expression as product of radical expressions.

103 Simplify

b. 81a4b4×2a Factor perfect square from radicand

81a4b4×2a Write radical expression as product of radical expressions

9a2b22a Simplify

Exercise 1.3.2

Simplify 50x2y3z

Answer

5|x||y|2yz

Notice the absolute value signs around x and y? That’s because their value must be positive!

Howto: Given the product of multiple radical expressions, use the product rule to combine them into one radical expression
  1. Express the product of multiple radical expressions as a single radical expression.
  2. Simplify.
Example 1.3.5: Using the Product Rule to Simplify the Product of Multiple Square Roots

Simplify the radical expression.

12×3

Solution

12×3Express the product as a single radical expression36Simplify6

Exercise 1.3.3

Simplify 50x×2x assuming x>0.

Answer

10|x|

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=52.

THE QUOTIENT RULE FOR SIMPLIFYING SQUARE ROOTS

The square root of the quotient ab is equal to the quotient of the square roots of a and b, where b0.

ab=ab

Howto: 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 1.3.6: Using the Quotient Rule to Simplify Square Roots

Simplify the radical expression.

536

Solution

536Write as quotient of two radical expressions56Simplify denominator

Exercise 1.3.4

Simplify 2x29y4

Answer

x23y2

We do not need the absolute value signs for y2 because that term will always be nonnegative.

Example 1.3.7: Using the Quotient Rule to Simplify an Expression with Two Square Roots

Simplify the radical expression.

234x11y26x7y

Solution

234x11y26x7yCombine numerator and denominator into one radical expression9x4Simplify fraction3x2Simplify square root

Exercise 1.3.5

Simplify 9a5b143a4b5

Answer

b43ab

Adding and Subtracting 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 and 32 is 42. However, it is often possible to simplify radical expressions, and that may change the radicand. The radical expression 18 can be written with a 2 in the radicand, as 32, so 2+18=2+32=42

Howto: 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​ 1.3.8: Adding Square Roots

Add 512+23.

Solution

We can rewrite 512 as 54×3. According the product rule, this becomes 543. The square root of 4 is 2, so the expression becomes 5×23, which is 103. Now we can the terms have the same radicand so we can add.

103+23=123

Exercise 1.3.6

Add 5+620

Answer

135

Example 1.3.9: Subtracting Square Roots

Subtract 2072a3b4c148a3b4c

Solution

Rewrite each term so they have equal radicands.

2072a3b4c=20942aa2(b2)2c=20(3)(2)|a|b22ac=120|a|b22ac

148a3b4c=1424aa2(b2)2c=14(2)|a|b22ac=28|a|b22ac

Now the terms have the same radicand so we can subtract.

120|a|b22ac28|a|b22ac=92|a|b22ac

Exercise 1.3.7

Subtract 380x445x

Answer

0

Rationalizing 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 bc, multiply by cc.

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+bc, then the conjugate is abc.

HowTo: 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 1.3.10: Rationalizing a Denominator Containing a Single Term

Write 23310 in simplest form.

Solution

The radical in the denominator is 10. So multiply the fraction by 1010. Then simplify.

23310×1010230303015

Exercise 1.3.8

Write 1232 in simplest form.

Answer

66

How to: Given an expression with a radical term and a constant in the denominator, rationalize the denominator
  1. Find the conjugate of the denominator.
  2. Multiply the numerator and denominator by the conjugate.
  3. Use the distributive property.
  4. Simplify.
Example 1.3.11: Rationalizing a Denominator Containing Two Terms

Write 41+5 in simplest form.

Solution

Begin by finding the conjugate of the denominator by writing the denominator and changing the sign. So the conjugate of 1+5 is 15. Then multiply the fraction by 1515.

41+5×15154454Use the distributive property51Simplify

Exercise 1.3.9

Write 72+3 in simplest form.

Answer

1473

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.

Understanding nth Roots

Suppose we know that a3=8. We want to find what number raised to the 3rd power is equal to 8. Since 23=8, we say that 2 is the cube root of 8.

The nth root of a is a number that, when raised to the nth power, gives a. For example, 3 is the 5th root of 243 because (3)5=243. If a is a real number with at least one nth root, then the principal nth root of a is the number with the same sign as a that, when raised to the nth power, equals a.

The principal nth root of a is written as na, where n is a positive integer greater than or equal to 2. In the radical expression, n is called the index of the radical.

Note: Principal nth Root

If a is a real number with at least one nth root, then the principal nth root of a, written as na, is the number with the same sign as a that, when raised to the nth power, equals a. The index of the radical is n.

Example 1.3.12: Simplifying nth Roots

Simplify each of the following:

  1. 532
  2. 44×410234
  3. 38x6125
  4. 843448
Solution

a. 532=2 because (2)5=32

b. First, express the product as a single radical expression. 44096=8 because 84=4096

c. 38x63125Write as quotient of two radical expressions2x25Simplify

d. 843243Simplify to get equal radicands643Add

Exercise 1.3.10

Simplify

  1. 3216
  2. 348045
  3. 639000+73576
Answer a

6

Answer b

6

Answer c

8839

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 n is even, then a cannot be negative.

a1n=na

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=(na)m=nam

All of the properties of exponents that we learned for integer exponents also hold for rational exponents.

Note: 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=(na)m=nam

Howto: 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 1.3.13: Writing Rational Exponents as Radicals

Write 34323 as a radical. Simplify.

Solution

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

34323=(3343)2=33432

We know that 3343=7 because 73=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=(3343)2=72=49

Exercise 1.3.11

Write 952 as a radical. Simplify.

Answer

(9)5=35=243

Example 1.3.14: Writing Radicals as Rational Exponents

Write 47a2 using a rational exponent.

Solution

The power is 2 and the root is 7, so the rational exponent will be 27. We get 4a27. Using properties of exponents, we get 47a2=4a27

Exercise 1.3.12

Write x(5y)9 using a rational exponent.

Answer

x(5y)92

Example 1.3.15: Simplifying Rational Exponents

Simplify:

a. 5(2x34)(3x15)

b. (169)12

Solution

a.

30x34x15Multiply the coefficients30x34+15Use properties of exponents30x1920Simplify

b.

(916)12Use definition of negative exponents916Rewrite as a radical916Use the quotient rule34Simplify

Exercise 1.3.13

Simplify (8x)13(14x65)

Answer

28x2315

Media

Access these online resources for additional instruction and practice with radicals and rational exponents.

Radicals

Rational Exponents

Simplify Radicals

Rationalize Denominator

Key Concepts

  • The principal square root of a number a is the nonnegative number that when multiplied by itself equals a. See Example.
  • If a and b are nonnegative, the square root of the product ab is equal to the product of the square roots of a and b See Example and Example.
  • If a and b are nonnegative, the square root of the quotient ab is equal to the quotient of the square roots of a and b See Example and Example.
  • We can add and subtract radical expressions if they have the same radicand and the same index. See Example and Example.
  • 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. See Example and Example.
  • The principal nth root of a is the number with the same sign as a that when raised to the nth power equals a. These roots have the same properties as square roots. See Example.
  • Radicals can be rewritten as rational exponents and rational exponents can be rewritten as radicals. See Example and Example.
  • The properties of exponents apply to rational exponents. See Example.

This page titled 1.3: Radicals and Rational Exponents is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

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