Beginning Algebra, 11th Edition

OBJECTIVE 4 Simplify radicals involving variables.Simplifying radicals with

variable radicands, such as , can get a little tricky.

If xrepresents a nonnegative number, then

If xrepresents a negative number, then the oppositeof x

(which is positive).

For example, but the oppositeof

This means that the square root of a squared number is always nonnegative. We can

use absolute value to express this.

252 = 5 , 21 - 522 = 225 = 5 , -5.

2 x^2 =-x,

2 x^2 =x.

2 x^2

508 CHAPTER 8 Roots and Radicals

NOTE A quick way to find the square root of a variable raised to an even power is to

divide the exponent by the index, 2. For example,

and.

6 , 2 = 3 10 , 2 = 5

2 x^6 =x^32 x^10 =x^5

For any real number a, 2 a^2 |a|.

2 a^2

The product and quotient rules apply when variables appear under radical sym-

bols, as long as the variables represent nonnegativereal numbers. To avoid negative

radicands, we assume variables under radical symbols are nonnegative in this text.

In such cases, absolute value bars are not necessary, since, for xÚ0, |x|=x.

NOW TRY
EXERCISE 7
Simplify each radical.
Assume that all variables
represent nonnegative real
numbers.

(a) (b)

(c)
B

13

t^2

, tZ 0

216 y^82 x^5

NOW TRY ANSWERS

7. (a) (b)

(c)

213

t

4 y^4 x^22 x

(e) ,

Quotient rule

=

25

x

=

25

2 x^2

xZ 0

B

5

x^2

Simplifying Radicals Involving Variables

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

(a) 2 x^4 =x^2 ,since 1 x^222 =x^4.

EXAMPLE 7

(c)

Factor; 4 is a perfect square.

Product rule

;

Commutative property

(d)

Product rule

=r^42 r 1 r^422 =r 8

= 2 r^8 # 2 r

= 2 r^8 #r

2 r^9

= 2 p^522

= 2 # 22 #p^524 = 21 p^522 =p^10

= 24 # 22 # 2 p^10

= 24 # 2 #p^10

28 p^10

(b)

Product rule

= 5 m^31 m^322 =m^6

= 225 # 2 m^6

225 m^6

NOW TRY

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