Dividing and Simplifying

Just as the root of a product can be expressed as the product of two roots, the root of a quotient can be expressed as the quotient of two roots.

Remember that an nth root is simplified when its radicand has no factors that are perfect nth powers. Recall too that we assume that no radicands represent negative quantities raised to an even power.

Example Simplify by taking roots of the numerator and denominator, assume all variables are positive.

Solution

4

43

3

49a)

54b)

x

m

n

4 4

49 49a)

x x

27

x

Taking the square roots of the numerator and denominator

34 43

3 3 3

54 54b)

m m

n n

3 3

3 3

27 2 m m

n

3 3 3

3 3

27 2m m

n

33 2m m

n

Solution

Example Divide and, if possible, simplify, assume all variables are positive.

4 53

3

48a)

3

24b)

2 3

x y

x

48 48a)

33

16

4

4 5 4 533

324 1 24

b) 2 32 3

x y x y

xx

3 5318

2x y

3 3 23 318

2x y y

2312

2xy y

23xy y

Rationalizing Denominators and Numerators (Part 1)*

When a radical expression appears in a denominator, it can be useful to find an equivalent expression in which the denominator no longer contains a radical. The procedure for finding such an expression is called rationalizing the denominator. We carry this out by multiplying by 1 in either of two ways.

One way is to multiply by a clever form of one under the radical to make the denominator of the radicand a perfect power.

5 5a)

7

7

7 7

35

49

35

49

Multiplying by clever form

of one under the radical

Example Rationalize each denominator.

35

7

3 3b)

25

33

15

5

3

3

3

15

5

3 15

5

Since the index is 3, we need 3 identical

factors in the denominator.

The denominator is now a perfect cube.

3 3

5 5

5

5

Solution

Example Rationalize each denominator.

23

5a)

33

b) 4

xy

xy

5 5a)

3 3x x

3

5

3

3

x

x

x

2

15

9

x

x

15

3

x

x

Multiplying by 1

23

232 23 3

3 3b)

4

2

24

y y

xy x

x y

x yy

23

3 33

3 2

8

y x y

x y

233 2

2

y x y

xy

233 2

2

x y

x

Another way to rationalize a denominator is to multiply by 1 outside the radical.

Sometimes in calculus it is necessary to rationalize a numerator. To do so, we multiply by a clever form of 1 to make the radicand in the numerator a perfect power.