§ 7.3

Multiplying and Simplifying Radical Expressions

Blitzer, Intermediate Algebra, 5e – Slide #2 Section 7.3

Multiplying Radicals

The Product Rule for RadicalsIf and are real numbers, then

The product of two nth roots is the nth root of the product.

n a n b

.nnn abba

Blitzer, Intermediate Algebra, 5e – Slide #3 Section 7.3

Multiplying Radicals

EXAMPLEEXAMPLE

Multiply: .55(b)45(a) 6 4633 xx

SOLUTIONSOLUTION

In each problem, the indices are the same. Thus, we multiply the radicals by multiplying the radicands.

3333 204545(a)

6 56 46 46 55555(b) xxxxx

Blitzer, Intermediate Algebra, 5e – Slide #4 Section 7.3

Multiplying Radicals

Check Point 1 on p 509Check Point 1 on p 509

Multiply: indices are the same. Note: problems from 1-19.

33 106(c)

115(a) 55

3 60

7 37 62(d) xx 7 412x

Blitzer, Intermediate Algebra, 5e – Slide #5 Section 7.3

Simplifying Radicals

EXAMPLEEXAMPLE

Simplify by factoring: .32(b)28(a) 3 32 yx

SOLUTIONSOLUTION

7428(a)

74

72

4 is the greatest perfect square that is a factor of 28.Take the square root of each factor.Write as 2.4

Blitzer, Intermediate Algebra, 5e – Slide #6 Section 7.3

Simplifying Radicals

3 23 3 48 xy Factor into two radicals.

.8 3y

CONTINUECONTINUEDD

3 242 xy Take the cube root of

3 233 32 4832(b) xyyx is the greatest perfect cube that is a factor of the radicand.

38y

Blitzer, Intermediate Algebra, 5e – Slide #7 Section 7.3

Simplifying Radicals

Check Point 2 on p 510 problems from 21-31Check Point 2 on p 510 problems from 21-31

80(a) 54516

3 40(b)3 58 3 52

4 32(c) 4 216 4 22

yx2200(d) yx 2100 2 yx 2||10

Blitzer, Intermediate Algebra, 5e – Slide #8 Section 7.3

Simplifying Radicals

EXAMPLEEXAMPLE

Simplify: .40 3x

SOLUTIONSOLUTION

We write the radicand as the product of the greatest perfect square factor and another factor. Because the index of the radical is 2, variables that have exponents that are divisible by 2 are part of the perfect square factor. We use the greatest exponents that are divisible by 2.

Use the greatest even power of each variable.

Group the perfect square factors.

xxx 23 10440

xx 104 2

xx 104 2 Factor into two radicals.

xx 102 Simplify the first radical.

Blitzer, Intermediate Algebra, 5e – Slide #9 Section 7.3

Simplifying Radicals

EXAMPLEEXAMPLE

Simplify: .964 11x

SOLUTIONSOLUTION

We write the radicand as the product of the greatest 4th power and another factor. Because the index is 4, variables that have exponents that are divisible by 4 are part of the perfect 4th factor. We use the greatest exponents that are divisible by 4.

Identify perfect 4th factors.

Group the perfect 4th factors.

Factor into two radicals.

4 384 11 61696 xxx

4 38 616 xx4 34 8 616 xx

Simplify the first radical.

4 32 62 xx

Blitzer, Intermediate Algebra, 5e – Slide #10 Section 7.3

Simplifying Radicals

Check Point 4 and 5 on p 512 problems from Check Point 4 and 5 on p 512 problems from 39-4739-47

3119(cp4) zyx xyzzyx 54xyzzyx 2108

3 141040(cp5) yx 3 2129 58 xyyx 3 243 52 xyyx

Blitzer, Intermediate Algebra, 5e – Slide #11 Section 7.3

Simplifying Radicals

Check Point 7 on p 513 problems from 61-77Check Point 7 on p 513 problems from 61-77

26(a) 3412

33 2 51610 (b) 3 3250

32

3 4850 3 4 250 3 4 100

4 364 2 84 )( yxyxc 4 4832 yx 4 48 216 yx 42 22 yx

DONE

Blitzer, Intermediate Algebra, 5e – Slide #13 Section 7.3

Simplifying Radicals

Simplifying Radical Expressions by FactoringA radical expression whose index is n is simplified when its radicand has no factors that are perfect nth powers. To simplify, use the following procedure:

1) Write the radicand as the product of two factors, one of which is the greatest perfect nth power.

2) Use the product rule to take the nth root of each factor.

3) Find the nth root of the perfect nth power.

Blitzer, Intermediate Algebra, 5e – Slide #14 Section 7.3

Simplifying Radicals

EXAMPLEEXAMPLE

If , express the function, f, in simplified form. 3 3248 xxf

SOLUTIONSOLUTION

Begin by factoring the radicand. There is no GCF.

This is the given function. 3 3248 xxf

Factor 48. 3 3286 x

Rewrite 8 as . 3 33 226 x32

Take the cube root of each factor. 3 33 33 226 x

Take the cube root of and 3 622 x 32 .2 3x

Blitzer, Intermediate Algebra, 5e – Slide #15 Section 7.3

Simplifying Radicals

Simplifying When Variables to Even Powers in a Radicand are Nonnegative Quantities

For any nonnegative real number a,

.aan n

Blitzer, Intermediate Algebra, 5e – Slide #16 Section 7.3

Multiplying Radicals

EXAMPLEEXAMPLE

Multiply and simplify:

.88(b)84(a) 5 895 3344 644 332 zxyzyxyzxzyx

SOLUTIONSOLUTION

Use the product rule.

Multiply.

Identify perfect 4th factors.

4 644 332 84(a) yzxzyx

4 94632 zyx4 8424216 zzyxx

4 64332 84 yzxzyx

Group the perfect 4th factors. 4 2844 216 zxzyx

Factor into two radicals.4 24 844 216 zxzyx

Blitzer, Intermediate Algebra, 5e – Slide #17 Section 7.3

Multiplying Radicals

5 895 334 88(b) zxyzyx

Use the product rule.

Multiply.

Identify perfect 5th factors.

Group the perfect 5th factors.

Factor into two radicals.

CONTINUECONTINUEDD

5 89334 88 zxyzyx

5 1112564 zyx

5 102105232 zzyyx

5 210105 232 zyzyx

5 25 10105 232 zyzyx

Simplify the first radical.

5 222 22 zyzxy

Factor into two radicals.4 22 22 zxxyz

Blitzer, Intermediate Algebra, 5e – Slide #18 Section 7.3

Simplifying Radicals

Important to Remember:

A radical expression of index n is not simplified if you can take any roots – that is if there are any factors of the radicand that are perfect nth powers.

Take all roots.