Algebra Demystified 2nd Ed

Chapter 5 e X P O N e N T S a N D r O O T S 113

DeMYSTiFieD / algebra DeMYSTiFieD / HuttenMuller / 000-0 / Chapter 5

✔SOLUTIONS

1.^44

2

5

x 25

y

= xy−

2.^23
1

xx 2 2312

x

()xx x

()

− ()()

+

=−+ −

3. x
   y

=xy−^1

4.^2
3

x 2 232

y

xy

()

= ()−

5.^23
25

x 23251

x

− xx

+

=−()()+ −

Roots

The square root of a number is the positive number whose square is the root.

For example, 3 is the square root of 9 because 3^2 = 9. It might seem that nega-

tive numbers could be square roots. It is true that ()−= 392 , but 9 is the sym-

bol for the nonnegative number whose square is 9. Sometimes we say that 3 is

the principal square root of 9.

EXAMPLES

16 = 4 because 4^2 =^16

81 = 9 because 9^2 =^81

In general, nab= if bn = a. If n is even, we assume b is the nonnegative root. In

this book, we assume even roots will be taken only of nonnegative numbers.

That is, if we are taking the square root, fourth root, sixth root, etc., we must

assume that the quantity under the radical symbol, n , is not negative. For

instance, in the expression x we assume that x is not negative. Note that there

is no problem with odd roots being negative numbers. For example^3 −=− 64 4

because (−4)^3 = (−4)(−4)(−4) = −64.

Root properties are similar to exponent properties.

Property 10 nnab= abb

Property 10 allows us to take the product followed by the root or we can take

the individual roots followed by the product.

EXAMPLES EXAMPLES