8.4 Radicals - Multiply and Divide Radicals

8.4

Radicals - Multiply and Divide Radicals

Objective: Multiply and divide radicals using the product and quotient rules of radicals.

Multiplying radicals is very simple if the index on all the radicals match. The prodcut rule of radicals which we have already been using can be generalized as follows:

Product Rule of Radicals: a m b ? c m d = ac m bd

Another way of stating this rule is we are allowed to multiply the factors outside the radical and we are allowed to multiply the factors inside the radicals, as long as the index matches. This is shown in the following example.

Example 1.

- 514 ? 46

- 20 84 - 204 ? 21 - 20 ? 221

- 40 21

Multiply outside and inside the radical Simplify the radical, divisible by 4 Take the square root where possible Multiply coefficients Our Solution

The same process works with higher roots

Example 2.

2 3 18 ? 6 315 12 3270

12 327 ? 10 12 ? 3 310

36 310

Multiply outside and inside the radical Simplify the radical, divisible by 27 Take cube root where possible Multiply coefficients Our Solution

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When multiplying with radicals we can still use the distributive property or FOIL just as we could with variables.

Example 3.

76(310 - 515) 2160 - 3590

214 ? 15 - 359 ? 10 21 ? 215 - 35 ? 310

4215 - 10510

Distribute, following rules for multiplying radicals Simplify each radical, finding perfect square factors Take square root where possible Multiply coefficients Our Solution

Example 4.

(5 - 23)(410 + 66) 450 + 630 - 830 - 1218 425 ? 2 + 630 - 830 - 129 ? 2 4 ? 52 + 630 - 830 - 12 ? 32 202 + 630 - 830 - 362

- 162 - 230

FOIL, following rules for multiplying radicals Simplify radicals, find perfect square factors Take square root where possible Multiply coefficients Combine like terms Our Solution

World View Note: Clay tablets have been discovered revealing much about Babylonian mathematics dating back from 1800 to 1600 BC. In one of the tables there is an approximation of 2 accurate to five decimal places (1.41421)

Example 5.

(25 - 36)(72 - 87)

1410 - 1635 - 2112 - 2442

14 10

-

16 35

-

214

?

3

-

24 42

14 10

-

16 35

-

21

?

23

-

24 42

1410 - 1635 - 423 - 2442

FOIL, following rules for multiplying radicals Simplify radicals, find perfect square factors Take square root where possible Multiply coefficient Our Solution

As we are multiplying we always look at our final solution to check if all the radi-

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cals are simplified and all like radicals or like terms have been combined.

Division with radicals is very similar to multiplication, if we think about division as reducing fractions, we can reduce the coefficients outside the radicals and reduce the values inside the radicals to get our final solution.

a m b a b Quotient Rule of Radicals: c m d = c m d

Example 6.

15 3108 20 32

Reduce

15 20

and

3108 2

by

dividing

by

5

and

2

respectively

3 354 4

3 327 ? 2 4

3 ? 3 32 4

Simplify radical, 54 is divisible by 27 Take the cube root of 27 Multiply coefficients

9 32 4

Our Solution

There is one catch to dividing with radicals, it is considered bad practice to have a radical in the denominator of our final answer. If there is a radical in the denominator we will rationalize it, or clear out any radicals in the denominator. We do this by multiplying the numerator and denominator by the same thing. The problems we will consider here will all have a monomial in the denominator. The way we clear a monomial radical in the denominator is to focus on the index. The index tells us how many of each factor we will need to clear the radical. For example, if the index is 4, we will need 4 of each factor to clear the radical. This is shown in the following examples.

Example 7.

6 5

Index is 2, we need two fives in denominator, need 1 more

3

6 5 5 5

Multiply numerator and denominator by 5

30 5

Our Solution

Example 8.

3 411 42

Index is 4, we need four twos in denominator, need 3 more

3 411 423 42 423

Multiply numerator and denominator by 423

3 488 2

Our Solution

Example 9.

4 32 7 325

The 25 can be written as 52. This will help us keep the numbers small

4 32 7 352

Index is 3, we need three fives in denominator, need 1 more

4 32 35 7 352 35

Multiply numerator and denominator by 35

4 310 7?5

4 310 35

Multiply out denominator Our Solution

The previous example could have been solved by multiplying numerator and denominator by 3252. However, this would have made the numbers very large and we would have needed to reduce our soultion at the end. This is why rewriting the radical as 352 and multiplying by just 35 was the better way to sim-

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plify.

We will also always want to reduce our fractions (inside and out of the radical) before we rationalize.

Example 10.

614 1222

7 211

Reduce coefficients and inside radical Index is 2, need two elevens, need 1 more

7 11

211

11

Multiply numerator and denominator by 11

77 2 ? 11

77 22

Multiply denominator Our Solution

The same process can be used to rationalize fractions with variables.

Example 11.

18 4

6x3

4

yz

8 4 10xy6z3

9 4 3x2 4 4 5y2z3

9 43x2

4

53

2

yz

4 4 5y2z3 4 53y2z

Reduce coefficients and inside radical

Index is 4. We need four of everything to rationalize, three more fives, two more ys and one more z.

Multiply numerator and denominator by 4 53y2z

94

375x2

2

yz

4 ? 5yz

94

375x2

2

yz

20 y z

Multiply denominator Our Solution

Beginning and Intermediate Algebra by Tyler Wallace is licensed under a Creative Commons Attribution 3.0 Unported License. ()

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