Multiplying and Simplifying Radicals

• The Product Rule for Radicals is given by:

Note that both of the radicals on the left have the same index.

n n na b a b

Throughout this section and the rest of the chapter we will assume that variables are nonnegative, and thus not need the absolute value sign.

• Example 1

Multiply the radicals

44 3 4

n n na b a b

4 3 4 4 12

5 2 252 3x xy 2 25 2 3x xy

3 25 6x y

• The Product Rule for Radicals can be used in another way. Rather than using it for multiplication, it can also be used to break up a radical expression.

n n na b a b

n n na b a b

This will help in simplifying radicals.

• Example 2

Simplify: 28

n n na b a b

Since the radical is a square root, we want to factor the radicand where one of the factors is a perfect square.

28 4 7 Now apply the rule.

4 7

2 7

• Example 3

Simplify: 3 272x y

n n na b a b

Since the radical is a square root, we want to factor the radicand where one of the factors is a perfect square.

3 272x y 2 236 2 x x y

n n na b a b

2 236 2 x x y

2 236 2x y x

Now apply the rule.

6 2xy x

• When factoring a radicand that includes powers of variables, try to get a variable factor with an exponent that is a multiple of the index n.

7 145 x ySimplify:

• Example 4

7 145 x y

5 2 10 45 x x y y

We have an index of 5, so we want exponents that are multiples of 5.

5 2 10 45 x x y y

5 10 2 45 5x y x y

n n na b a b

Now apply the rule.

2 2 45xy x y

55 2 2 455 x y x y

510 2 25 5y y y

In this last problem we used …

In the future, we will skip the middle step, and use the shortcut of dividing the exponent by the index, leaving out the middle step.

• Example 5

Simplify: 3 73 72x y

n n na b a b

Since the radical is a cube root, we want to factor the radicand where one of the factors is a perfect cube.

3 73 72x y 3 63 8 9 x y y

3 63 38 9x y y 2 32 9xy y

• Example 6

Multiply and simplify: 3 7 84 48 4x y xy

Multiply using the product rule.

3 7 84 48 4x y xy 3 7 84 8 4x y xy

4 154 32x y

Since the radical is a fourth root, we want to factor the radicand where one of the factors is a perfect fourth power.

4 154 32x y

4 12 34 16 2x y y 4 12 34 416 2x y y

2 342 2xy y