Multiplying, Dividing, and Simplifying Radicals

Multiply square root radicals.

Simplify radicals by using the product rule.

Simplify radicals by using the quotient rule.

Simplify radicals involving variables.

Simplify other roots.

8.2

2

3

4

5

1

Product Rule for Radicals

For nonnegative real numbers a and b,

and

That is, the product of two square roots is the square root of the product, and the square root of a product is the product of the square roots.

Multiply square root radicals.

a b a b .a b a b

It is important to note that the radicands not be negative numbers in the product rule. Also, in general, .x y x y

6 11

Solution:

Find each product. Assume that 0.x

3 5

6 11

13 x

10 10

3 5

13 x

10 10

15

66

13x

100 10

EXAMPLE 1 Using the Product Rule to Multiply Radicals

Simplify radicals by using the product rule.

A square root radical is simplified when no perfect square factor remains under the radical sign.

This can be accomplished by using the product rule:

a b a b

Simplify each radical.

500

Solution:

60

17

4 15

100 5

It cannot be simplified further.

2 15

10 5

EXAMPLE 2 Using the Product Rule to Simplify Radicals

Find each product and simplify.

6 2

Solution:

10 50

6 2

10 50 500 100 5 10 5

12 2 3

EXAMPLE 3 Multiplying and Simplifying Radicals

12 5 2

12 25 2 3 5 4 10 3 4 5 10 12 50 12 25 2

60 2

The quotient rule for radicals is similar to the product rule.

Simplify radicals by using the quotient rule.

Simplify each radical.

48

3

Solution:

4

49

5

36

4

49

2

7

48

3 16 4

5

36

5

6

EXAMPLE 4 Using the Quotient Rule to Simplify Radicals

Simplify.

Solution:

8 50

4 5

8 50

4 5

502

5 2 10 2 10

EXAMPLE 5 Using the Quotient Rule to Divide Radicals

Simplify.

Solution:

3 7

8 2

3 7

8 2

21

16

21

16

21

4

EXAMPLE 6 Using Both the Product and Quotient Rules

Simplify radicals involving variables.

Radicals can also involve variables.

The square root of a squared number is always nonnegative. The absolute value is used to express this.

The product and quotient rules apply when variables appear under the radical sign, as long as the variables represent only nonnegative real numbers

2For any real number , .a a a

0, .x x x

Simplify each radical. Assume that all variables represent positive real numbers.

Solution:

6x

8100p

4

7

y

3x 23 6Since x x

8100 p 410p

4

7

y

2

7

y

EXAMPLE 7 Simplifying Radicals Involving Variables

Finding the square root of a number is the inverse of squaring a number. In a similar way, there are inverses to finding the cube of a number or to finding the fourth or greater power of a number.

The nth root of a is written

Find cube, fourth, and other roots.

.n a

In the number n is the index or order of the radical.,n a

n a

Radical sign

IndexRadicand

It can be helpful to complete and keep a list to refer to of third and fourth powers from 1-10.

Properties of Radicals

For all real number for which the indicated roots exist,

Simplify other roots.

To simplify cube roots, look for factors that are perfect cubes. A perfect cube is a number with a rational cube root.

For example, , and because 4 is a rational number, 64 is a perfect cube.

3 64 4

nand . 0n

n n n

n

a aa b ab b

bb

Find each cube root.

3 64

3 27

3 512

4

3

EXAMPLE 9 Finding Cube Roots

Solution:

8

Simplify each radical.

Solution:

3 108

4 160

416

625

33 27 4 33 4

4 16 10 4 416 10 42 10

4

4

16

625

2

5

EXAMPLE 8 Simplifying Other Roots

Find each root.

4 81

4 81

4 81

5 243

5 243

3

3

Not a real number.

3

3

Solution:

EXAMPLE 10 Finding Other Roots

Simplify each radical.

Solution:

3 9z

3 68x

3 554t

15

3a

64

3z

22x3 63 8 x

3 3 227 2t t 3 33 227 2t t 3 23 2t t

3 15

3 64

a

5

4

a

EXAMPLE 9 Simplifying Cube Roots Involving Variables

Adding and Subtracting Radicals

Add and subtract radicals.

Simplify radical sums and differences.

Simplify more complicated radical expressions.

8.3

2

3

1

Add and subtract radicals.

We add or subtract radicals by using the distributive property. For example,

8 3 368 6 3

.14 3

and 52 2 3,

32 3as well as and 2 3 .

Radicands are different

Indexes are different

Only like radicals — those which are multiples of the same root of the same number — can be combined this way. The preceding example shows like radicals. By contrast, examples of unlike radicals are

Note that cannot be simplified.35 + 5

Add or subtract, as indicated.

Solution:

8 5 2 5 3 11 12 11 7 10

8 2 5

10 5

3 12 11

9 11

It cannot be added by the distributive property.

EXAMPLE 1 Adding and Subtracting Like Radicals

Simplify radical sums and differences.

Sometimes, one or more radical expressions in a sum or difference must be simplified. Then, any like radicals that result can be added or subtracted.

Add or subtract, as indicated.

Solution:

27 12 5 200 6 18332 54 4 2

3 3 2 3

5 3

5 100 2 6 9 2

5 100 2 6 9 2

50 2 18 2

32 2

3 332 27 2 4 2

3 32 3 2 4 2

3 36 2 4 2

310 2

EXAMPLE 2 Simplifying Radicals to Add or Subtract

Simplify more complicated radical expressions.

When simplifying more complicated radical expressions, recall the rules for order of operations.

A sum or difference of radicals can be simplified only if the radicals are like radicals. Thus, cannot be simplified further.

5 3 5 4 5, but 5 5 3

Simplify each radical expression. Assume that all variables represent nonnegative real numbers.

7 21 2 27

7 21 2 27

147 2 27

49 3 2 27

49 3 2 27

7 3 2 27

7 3 2 3 3

7 3 6 3

13 3

6 3 8r r

6 2 2r r

6 3 2 2r r

18 2 2r r

9 2 2 2r r

3 2 2 2r r

5 2r

Solution:

EXAMPLE 3 Simplifying Radical Expressions

Simplify each radical expression. Assume that all variables represent nonnegative real numbers.

2y 72 18y

29 8 9 2y y

23 8 3 2y y

23 2 2 3 2y y

26 2 3 2y y

6 2 3 2y y

3 2y

3 2y

3 33 3 5 2 3x x x x

3 34 481 5 24x x

3 33 33 327 3 5 8 3x x x x

3 33 3 10 3x x x x

313 3x x

Solution:

EXAMPLE 3 Simplifying Radical Expressions (cont’d)

Rationalizing the Denominator

Rationalize denominators with square roots.

Write radicals in simplified form.

Rationalize denominators with cube roots.

8.4

2

3

1

Rationalize denominators with square roots.

It is easier to work with a radical expression if the denominators do not contain any radicals.

1 1 2

22 2

2

2

2.

2

This process of changing the denominator from a radical, or irrational number, to a rational number is called rationalizing the denominator.

The value of the radical expression is not changed; only the form is changed, because the expression has been multiplied by 1 in the form of

Rationalize each denominator.

Solution:

18

24618

2 6 6 18 6

2 6

18 6

12

16

8

216

2 2 2 16 2

2 2

16 2

4 4 2

3 6

2

EXAMPLE 1 Rationalizing Denominators

Write radicals in simplified form.

Conditions for Simplified Form of a Radical

1. The radicand contains no factor (except 1) that is a perfect square (in dealing with square roots), a perfect cube (in dealing with cube roots), and so on.

2. The radicand has no fractions.

3. No denominator contains a radical.

Solution:

5.

18

5

18

8

5

18

18

1 5 18

18

5 9 2

18

5 9 2

18

3 5 2

18

3 10

18

10

6

EXAMPLE 2 Simplifying a Radical

Simplify

Simplify

Solution:

1 5.

2 6

1 5

2 6

5

12

5

12

35

2 3 3

5 3

6

15

6

EXAMPLE 3 Simplifying a Product of Radicals

Simplify. Assume that p and q are positive numbers.

Solution:

5p

q

5 qp

q q

5pq

q

EXAMPLE 4 Simplifying Quotients Involving Radicals

35

7

pq

2 235

7

p q

2 25

7

7

7

p q

2 2 35

7

p q

2 25

7

p q

2 25

7

p q

Rationalize each denominator.

Solution:

35

6

3

3

2

3

3

3

3, 0

4xx

2

3

3

23

3 65

6 6

3 2

3 3

5 6

6

3 180

6

2

3

3

23

3 32

3 3

3 2

3 3

2 3

3

3 18

3

3 2 2

3 23 2

3 4

4

3

4

x

x x

3 2

3 3 3

3 16

4

x

x

23 3 2 8

4

x

x

3 23 8 6

4

x

x

3 26

2

x

x

EXAMPLE 5 Rationalizing Denominators with Cube Roots

More Simplifying and Operations with Radicals

Simplify products of radical expressions.

Use conjugates to rationalize denominators of radical expressions.

Write radical expressions with quotients in lowest terms.

8.5

2

3

1

Find each product and simplify.

Solution:

2 8 20 2 5 3 3 2 2

2 2 2 4 5

2 2 2 4 5

2 2 2 2 5

4 2 5 2

4 2 10

2 3 2 2 2 5 3 3 5 3 2 2

6 11 10 6

11 9 6

EXAMPLE 1 Multiplying Radical Expressions

Find each product and simplify.

Solution:

2 5 10 2

2 10 2 2 5 10 5 2

20 2 50 10

2 5 2 5 2 10

EXAMPLE 1 Multiplying Radical Expressions (cont’d)

Find each product. Assume that x ≥ 0.

Solution:

2

5 3 2

4 2 5 2

2 x

2

25 2 5 3 3 2

24 2 2 4 2 5 5 222 2 2 x x

5 6 5 9

14 6 5

32 40 2 25

57 40 2

4 4 x x

Remember only like radicals can be combined!

EXAMPLE 2 Using Special Products with Radicals

Using a Special Product with Radicals.

Example 3 uses the rule for the product of the sum and difference of two terms,

2 2.x y x y x y

Find each product. Assume that 0.y

Solution:

3 2 3 2 4 4y y

2 2

3 2

3 4

1

2 2

4y

16y

EXAMPLE 3 Using a Special Product with Radicals

The results in the previous example do not contain radicals. The pairs being multiplied are called conjugates of each other. Conjugates can be used to rationalize the denominators in more complicated quotients, such as

Use conjugates to rationalize denominators of radical expressions.

2.

4 3

Using Conjugates to Rationalize a Binomial Denominator

To rationalize a binomial denominator, where at least one of those terms is a square root radical, multiply numerator and denominator by the conjugate of the denominator.

Simplify by rationalizing each denominator. Assume that 0.t 3

2 55+3

2 5

2 5

2 55 2

3

2

2

3 2 5

2 5

3 2 5

4 5

3 2 5

1

3 2 5

2 5

2 5

5 3

2 5

2 2

2 5 5 6 3 5

2 5

5 5 11

4 5

5 5 11

1

5 5 11

11 5 5

Solution:

EXAMPLE 4 Using Conjugates to Rationalize Denominators

Simplify by rationalizing each denominator. Assume that 0.t

3

2 t

23

2 2

t

tt

2

2

3 2

2

t

t

3 2

4

t

t

Solution:

EXAMPLE 4 Using Conjugates to Rationalize Denominators (cont’d)

Write in lowest terms.

Solution:

5 3 15

10

5 3 3

10

3 3

2

EXAMPLE 5 Writing a Radical Quotient in Lowest Terms

Using Rational Numbers as Exponents

Define and use expressions of the form a1/n.

Define and use expressions of the form am/n.

Apply the rules for exponents using rational exponents.

Use rational exponents to simplify radicals.

8.7

2

3

4

1

Define and use expressions of the form a1/n.

Now consider how an expression such as 51/2 should be defined, so that all the rules for exponents developed earlier still apply. We define 51/2 so that

51/2 · 51/2 = 51/2 + 1/2 = 51 = 5.

This agrees with the product rule for exponents from Section 5.1. By definition,

Since both 51/2 · 51/2 and equal 5,

this would seem to suggest that 51/2 should equal

Similarly, then 51/3 should equal

5 5 5.

5 5

3 5.5.

Review the basic rules for exponents:

m n m na a a m

m nn

aa

a nm mna a

Slide 8.7-4

a1/nIf a is a nonnegative number and n is a positive integer, then

1/ .n na a

Slide 8.7-5

Define and use expressions of the form a1/n.

Notice that the denominator of the rational exponent is the index of the radical.

Simplify.

491/2

10001/3

811/4

Solution:

49 7

3 1000 10

4 81 3

Slide 8.7-6

EXAMPLE 1 Using the Definition of a1/n

Define and use expressions of the form am/n.

Now we can define a more general exponential expression, such as 163/4. By the power rule, (am)n = amn, so

333/ 4 1/ 4 3416 16 16 2 8.

However, 163/4 can also be written as

Either way, the answer is the same. Taking the root first involves smaller numbers and is often easier. This example suggests the following definition for a m/n.

1/ 4 1/ 43/ 4 3 416 16 4096 4096 8.

am/nIf a is a nonnegative number and m and n are integers with n > 0, then

/ 1/ .mmm n n na a a

Slide 8.7-8

Evaluate.

95/2

85/3

–272/3

Solution:

51/ 29 53

51/38 52

21/327 9

243

32

23

Slide 8.7-9

EXAMPLE 2 Using the Definition of am/n

Earlier, a–n was defined as

for nonzero numbers a and integers n. This same result applies to negative rational exponents.

Using the definition of am/n.

1nn

aa

a−m/nIf a is a positive number and m and n are integers, with n > 0, then

//

1.m n

m na

a

A common mistake is to write 27–4/3 as –273/4. This is incorrect. The negative exponent does not indicate a negative number. Also, the negative exponent indicates to use the reciprocal of the base, not the reciprocal of the exponent.

Slide 8.7-10

Solution:

Evaluate.

36–3/2

81–3/4

3/ 2

1

36

3/ 4

1

81

31/ 2

1

36

3

1

6

1

216

31/ 4

1

81

3

1

3

1

27

Slide 8.7-11

EXAMPLE 3 Using the Definition of a−m/n

Apply the rules for exponents using rational exponents.

All the rules for exponents given earlier still hold when the exponents are fractions.

Slide 8.7-13

Solution:

Simplify. Write each answer in exponential form with only positive exponents.

1/3 2/37 72/3

1/3

9

9

5/327

8

1/ 2 2

5/ 2

3 3

3

1/3 2 /37 7

2/3 1/39 9

5/3

5/3

27

8

51/3

51/3

27

8

5

5

3

2

1/ 2 4/ 2 5/ 23 2/ 23 3

Slide 8.7-14

EXAMPLE 4 Using the Rules for Exponents with Fractional Exponents

Simplify. Write each answer in exponential form with only positive exponents. Assume that all variables represent positive numbers.

Solution:

62/3 1/3 2a b c

2/3 1/3

1

r r

r

32/3

1/ 4

a

b

6 6 62/3 1/3 2a b c 12/3 6/3 12a b c 4 2 12a b c

2/3 1/3 3/3r 6/3r 2r

32/3

31/ 4

a

b

6/3

3/ 4

a

b

2

3/ 4

a

b

Slide 8.7-15

EXAMPLE 5 Using Fractional Exponents with Variables

Use rational exponents to simplify radicals.

Sometimes it is easier to simplify a radical by first writing it in exponential form.

Slide 8.7-17

Simplify each radical by first writing it in exponential form.

4 212

36 x

1/ 4212 1/ 212 12

1/ 63x 1/ 2x 0x x

Solution:

2 3

Slide 8.7-18

EXAMPLE 6 Simplifying Radicals by Using Rational Exponents

Solving Equations with Radicals

Solve radical equations having square root radicals.

Identify equations with no solutions.

Solve equations by squaring a binomial.

Solve radical equations having cube root radicals.

8.6

2

3

4

1

Solving Equations with Radicals.

A radical equation is an equation having a variable in the radicand, such as

1 3x or 3 8 9x x

To solve radical equations having square root radicals, we need a new property, called the squaring property of equality.

Be very careful with the squaring property: Using this property can give a new equation with more solutions than the original equation has. Because of this possibility, checking is an essential part of the process. All proposed solutions from the squared equation must be checked in the original equation.

Solve radical equations having square root radicals.

Squaring Property of Equality

If each side of a given equation is squared, then all solutions of the original equation are among the solutions of the squared equation.

Solve.

Solution:

It is important to note that even though the algebraic work may be done perfectly, the answer produced may not make the original equation true.

9 4x

229 4x

9 16x 9 169 9x

7x 7x 7

EXAMPLE 1 Using the Squaring Property of Equality

Solve.

Solution:

3 9 2x x

2 2

3 9 2x x

3 9 4x x

3 33 9 4xx x x

9x

9

EXAMPLE 2 Using the Squaring Property with a Radical on Each Side

Solution:

Solve.

4x

2 2

4x

16x

16 44 4

4x

False

Because represents the principal or nonnegative square root of x in Example 3, we might have seen immediately that there is no solution.

x

Check:

EXAMPLE 3 Using the Squaring Property When One Side Is Negative

Solving a Radical EquationStep 1 Isolate a radical. Arrange the terms so that a radical is

isolated on one side of the equation.

Solving a Radical Equation.

Step 6 Check all proposed solutions in the original equation.

Step 5 Solve the equation. Find all proposed solutions.

Step 4 Repeat Steps 1-3 if there is still a term with a radical.

Step 3 Combine like terms.

Step 2 Square both sides.

Solution:

Solve 2 4 16.x x x

22 2 4 16x x x

2 22 24 16x xx x x

44 40 16xx x 4 1

4 4

6x

4x

Since x must be a positive number the solution set is Ø.

EXAMPLE 4 Using the Squaring Property with a Quadratic Expression

Solve

Solution:

2 1 10 9.x x

222 1 10 9x x

2 10 94 4 1 10 99 10x x xx x 24 14 8 0x x

2 1 2 8 0x x

2 8 0x 2 1 0x 4x 1

2x

Since 2x-1 must be positive the solution set is {4}.

or

EXAMPLE 5 Using the Squaring Property when One Side Has Two Terms

Solve.

Solution:

25 6x x

625 66x x

2 2

25 6x x 225 12 325 256x x xx x 20 13 36x x

0 4 9x x 0 9x 0 4x

9x 4x

The solution set is {4,9}.

or

EXAMPLE 6 Rewriting an Equation before Using the Squaring Property

Solve equations by squaring a binomial.

Errors often occur when both sides of an equation are squared. For instance, when both sides of

are squared, the entire binomial 2x + 1 must be squared to get 4x2 + 4x + 1. It is incorrect to square the 2x and the 1 separately to get 4x2 + 1.

9 2 1x x

Solve.

Solution:

1 4 1x x

1 1 4x x

2 2

1 1 4x x

1 1 2 4 4x x x

224 2 4x

16 4 16x 32

4 4

4x

8x The solution set is {8}.

EXAMPLE 7 Using the Squaring Property Twice

Solve radical equations having cube root radicals.

We can extend the concept of raising both sides of an equation to a power in order to solve radical equations with cube roots.

Solve each equation.

Solution:

3 37 4 2x x 3 2 3 26 27x x

3 3

3 2 3 26 27x x 2 26 27x x

20 26 27x 0 27 1x x

0 27x 0 1x 27x 1x

3 33 37 4 2x x

7 4 2x x 3 2

3 3

x

2

3x

2

3

27,1

or

EXAMPLE 8 Solving Equations with Cube Root Radicals