Radicals and Rational Exponents

Learning Objectives

In this section, you will:

• 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, and use the Pythagorean Theorem.

Figure 1

a2+b252+122169===c2c2c2𝑎2+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=16,42=16, the square root of 1616 is 4.4. 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 a𝑎 is a positive real number, then the square root of a𝑎 is a number that, when multiplied by itself, gives a.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.𝑎. The square root obtained using a calculator is the principal square root.

The principal square root of a𝑎 is written as a−−√.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.

PRINCIPAL SQUARE ROOT

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

Q&A

Does 25−−√=±5?25=±5?

No. Although both 5252 and (−5)2(−5)2 are 25,25, the radical symbol implies only a nonnegative root, the principal square root. The principal square root of 25 is 25−−√=5.25=5.

EXAMPLE 1

Evaluating Square Roots

Evaluate each expression.

1. ⓐ 100−−−√100
2. ⓑ 16−−√−−−−√16
3. ⓒ 25+144−−−−−−−√25+144
4. ⓓ 49−−√−81−−√49−81

Solution

1. ⓐ 100−−−√=10100=10 because 102=100102=100
2. ⓑ 16−−√−−−−√=4–√=216=4=2 because 42=1642=16 and 22=422=4
3. ⓒ 25+144−−−−−−−√=169−−−√=1325+144=169=13 because 132=169132=169
4. ⓓ 49−−√−81−−√=7−9=−249−81=7−9=−2 because 72=4972=49 and 92=8192=81

Q&A

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

No. 25−−√+144−−−√=5+12=17.25+144=5+12=17. This is not equivalent to 25+144−−−−−−−√=13.25+144=13. The order of operations requires us to add the terms in the radicand before finding the square root.

TRY IT #1

Evaluate each expression.

1. ⓐ 225−−−√225
2. ⓑ 81−−√−−−−√81
3. ⓒ 25−9−−−−−√25−9
4. ⓓ 36−−√+121−−−√36+121

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−−√15 as 3–√⋅5–√.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 a𝑎 and b𝑏 are nonnegative, the square root of the product ab𝑎𝑏 is equal to the product of the square roots of a𝑎 and b.𝑏.

ab−−√=a−−√⋅b√𝑎𝑏=𝑎⋅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 2

Using the Product Rule to Simplify Square Roots

Simplify the radical expression.

1. ⓐ 300−−−√300
2. ⓑ 162a5b4−−−−−−√162𝑎5𝑏4

Solution

1. ⓐ
   100⋅3−−−−−√100−−−√⋅3–√103–√Factor perfect square from radicand.Write radical expression as product of radical expressions.Simplify.100⋅3Factor perfect square from radicand.100⋅3Write radical expression as product of radical expressions.103Simplify.
2. ⓑ
   81a4b4⋅2a−−−−−−−−−√81a4b4−−−−−√⋅2a−−√9a2b22a−−√Factor perfect square from radicand.Write radical expression as product of radical expressions.Simplify.81𝑎4𝑏4⋅2𝑎Factor perfect square from radicand.81𝑎4𝑏4⋅2𝑎Write radical expression as product of radical expressions.9𝑎2𝑏22𝑎Simplify.

TRY IT #2

Simplify 50x2y3z−−−−−−−√.50𝑥2𝑦3𝑧.

HOW TO

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 3

Using the Product Rule to Simplify the Product of Multiple Square Roots

Simplify the radical expression.
12−−√⋅3–√12⋅3

Solution

12⋅3−−−−√36−−√6Express the product as a single radical expression.Simplify.12⋅3Express the product as a single radical expression.36Simplify.6

TRY IT #3

Simplify 50x−−−√⋅2x−−√50𝑥⋅2𝑥 assuming x>0.𝑥>0.

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 as 5√2√.52.

THE QUOTIENT RULE FOR SIMPLIFYING SQUARE ROOTS

The square root of the quotient ab𝑎𝑏 is equal to the quotient of the square roots of a𝑎 and b,𝑏, where b≠0.𝑏≠0.

ab−−√=a−−√b√𝑎𝑏=𝑎𝑏

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 4

Using the Quotient Rule to Simplify Square Roots

Simplify the radical expression.

536−−√536

Solution

5√36√5√6Write as quotient of two radical expressions.Simplify denominator.536Write as quotient of two radical expressions.56Simplify denominator.

TRY IT #4

Simplify 2x29y4−−−√.2𝑥29𝑦4.

EXAMPLE 5

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

Simplify the radical expression.

234x11y√26x7y√234𝑥11𝑦26𝑥7𝑦

Solution

234x11y26x7y−−−−−√9×4−−−√3×2 Combine numerator and denominator into one radical expression.Simplify fraction.Simplify square root.234𝑥11𝑦26𝑥7𝑦Combine numerator and denominator into one radical expression.9𝑥4Simplify fraction.3𝑥2 Simplify square root.

TRY IT #5

Simplify 9a5b14√3a4b5√.9𝑎5𝑏143𝑎4𝑏5.

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–√2 and 32–√32 is 42–√.42. However, it is often possible to simplify radical expressions, and that may change the radicand. The radical expression 18−−√18 can be written with a 22 in the radicand, as 32–√,32, so 2–√+18−−√=2–√+32–√=42–√.2+18=2+32=42.

HOW TO

Given a radical expression requiring addition or subtraction of square roots, simplify.

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

EXAMPLE 6

Adding Square Roots

Add 512−−√+23–√.512+23.

Solution

We can rewrite 512−−√512 as 54⋅3−−−√.54·3. According the product rule, this becomes 54–√3–√.543. The square root of 4–√4 is 2, so the expression becomes 5(2)3–√,5(2)3, which is 103–√.103. Now the terms have the same radicand so we can add.

103–√+23–√=123–√103+23=123

TRY IT #6

Add 5–√+620−−√.5+620.

EXAMPLE 7

Subtracting Square Roots

Subtract 2072a3b4c−−−−−−√−148a3b4c−−−−−√.2072𝑎3𝑏4𝑐−148𝑎3𝑏4𝑐.

Solution

Factor 9 out of the first term so that both terms have equal radicands.

2072a3b4c−−−−−−√==209⋅8a3b4c−−−−−−−√20(3)8a3b4c−−−−−√==209–√8a3b4c−−−−−√608a3b4c−−−−−√2072𝑎3𝑏4𝑐=209⋅8𝑎3𝑏4𝑐=2098𝑎3𝑏4𝑐=20(3)8𝑎3𝑏4𝑐=608𝑎3𝑏4𝑐

So

2072a3b4c−−−−−−√−148a3b4c−−−−−√=608a3b4c−−−−−√−148a3b4c−−−−−√=468a3b4c−−−−−√2072𝑎3𝑏4𝑐−148𝑎3𝑏4𝑐=608𝑎3𝑏4𝑐−148𝑎3𝑏4𝑐=468𝑎3𝑏4𝑐

TRY IT #7

Subtract 380x−−−√−445x−−−√.380𝑥−445𝑥.

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√,𝑏𝑐, multiply by c√c√.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√,𝑎+𝑏𝑐, then the conjugate is a−bc√.𝑎−𝑏𝑐.

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 8

Rationalizing a Denominator Containing a Single Term

Write 23√310√23310 in simplest form.

Solution

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

23√310√⋅10√10√   230√30          30√1523310⋅1010   23030          3015

TRY IT #8

Write 123√2√1232 in simplest form.

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 9

Rationalizing a Denominator Containing Two Terms

Write 41+5√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–√1+5 is 1−5–√.1−5. Then multiply the fraction by 1−5√1−5√.1−51−5.

41+5√⋅1−5√1−5√4−45√−45–√−1Use the distributive property.Simplify.41+5⋅1−51−54−45−4Use the distributive property.5−1Simplify.

TRY IT #9

Write 72+3√72+3 in simplest form.

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.𝑎3=8. We want to find what number raised to the 3rd power is equal to 8. Since 23=8,23=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 a.𝑎. For example, −3−3 is the 5th root of −243−243 because (−3)5=−243.(−3)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 a.𝑎.

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

PRINCIPAL n𝑛 TH ROOT

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

EXAMPLE 10

Simplifying nth Roots

Simplify each of the following:

1. ⓐ −32−−−−√5−325
2. ⓑ 4–√4⋅1,024−−−−−√444⋅1,0244
3. ⓒ −8×6125−−−√3−8𝑥61253
4. ⓓ 83–√4−48−−√4834−484

Solution

1. ⓐ −32−−−−√5=−2−325=−2 because (−2)5=−32(−2)5=−32
2. ⓑFirst, express the product as a single radical expression. 4,096−−−−√4=84,0964=8 because 84=4,09684=4,096
3. ⓒ −8×6√3125√3−2x25Write as quotient of two radical expressions.Simplify.−8𝑥631253Write as quotient of two radical expressions.−2𝑥25Simplify.
4. ⓓ 83–√4−23–√463–√4 Simplify to get equal radicands.Add.834−234Simplify to get equal radicands.634 Add.

TRY IT #10

Simplify.

1. ⓐ −216−−−−√3−2163
2. ⓑ 380√45√4380454
3. ⓒ 69,000−−−−−√3+7576−−−√369,0003+75763

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=a−−√n𝑎1𝑛=𝑎𝑛

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=(a−−√n)m=am−−−√n𝑎𝑚𝑛=(𝑎𝑛)𝑚=𝑎𝑚𝑛

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

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=(a−−√n)m=am−−−√n𝑎𝑚𝑛=(𝑎𝑛)𝑚=𝑎𝑚𝑛

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 11

Writing Rational Exponents as Radicals

Write 3432334323 as a radical. Simplify.

Solution

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

34323=(343−−−√3)2=3432−−−−√334323=(3433)2=34323

We know that 343−−−√3=73433=7 because 73=343.73=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=(343−−−√3)2=72=4934323=(3433)2=72=49

TRY IT #11

Write 952952 as a radical. Simplify.

EXAMPLE 12

Writing Radicals as Rational Exponents

Write 4a2√74𝑎27 using a rational exponent.

Solution

The power is 2 and the root is 7, so the rational exponent will be 27.27. We get 4a27.4𝑎27. Using properties of exponents, we get 4a2√7=4a−27.4𝑎27=4𝑎−27.

TRY IT #12

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

EXAMPLE 13

Simplifying Rational Exponents

Simplify:

1. ⓐ 5(2×34)(3×15)5(2𝑥34)(3𝑥15)
2. ⓑ (169)−12(169)−12

Solution

ⓐ
30x34x1530x34+1530x1920Multiply the coefficients.Use properties of exponents.Simplify.30𝑥34𝑥15Multiply the coefficients.30𝑥34+15Use properties of exponents.30𝑥1920Simplify.

ⓑ
(916)12916−−√9√16√34  Use definition of negative exponents.  Rewrite as a radical.  Use the quotient rule.  Simplify.(916)12  Use definition of negative exponents.916  Rewrite as a radical.916  Use the quotient rule.34  Simplify.

TRY IT #13

Simplify (8x)13(14×65).