Easy Algebra Step-by-Step

54 Easy Algebra Step-by-Step

Step 2. − 16 is negative and 4 is even, so ()−

14 /

is

not a real number.

()−^14 / =−^416 is not defi ned for real

numbers.

d. ⎛−

⎝⎜

⎛⎛

⎝⎝

⎞

⎠⎟

⎞⎞

⎠⎠

32

243

15 /

Step 1. Apply xx1/n n.

⎛−

⎝

⎛⎛⎛⎛

⎝⎝

⎛⎛⎛⎛ ⎞

⎠

⎞⎞⎞

⎠⎠

(^32) ⎞⎞⎞⎞ =−
243

32

243

15

5

/

Step 2. Find the principal fi fth root of −

32

243

.

⎝⎝⎛⎛⎛⎝⎛⎛⎛⎛⎛⎛− ⎞⎠⎞⎞⎞⎠⎠⎞⎞⎞⎞ =− =−

32

243

32

243

2

3

15

5

/

Rational Exponents

If x is a real number and m and n are natural numbers, then

(a) xm//n ()xx/nmn or (b) xm//n ()xxmm n, provided that in all cases

even roots of negative numbers do not occur.

When you evaluate the exponential expression xmn/, you can fi nd the

nth root of x fi rst and then raise the result to the mth power, or you can

raise x to the mth power fi rst and then fi nd the nth root of the result. For

most numerical situations, you usually will fi nd it easier to fi nd the root fi rst

and then raise to the power (as you will observe from the sample problems

shown here).

Problem Evaluate using xm//n ()x/nm.

a. ⎛⎝⎜⎛⎛⎝⎝− ⎞⎠⎟⎞⎞⎠⎠

1

8

23 /

b. ()^32 //

P

())^1 ////^4 ≠− 2.

− 2222 ⋅− ⋅− ⋅− = 16 , not − 16.

When you evaluate exponential

expressions that have unit fraction

exponents, you should practice

doing Step 1 mentally. For instance,

4912 / = 7 , ()))^13 //// =−^2 ,^

1

32

1

2

⎛^15

⎝⎜

⎛⎛

⎝⎝

⎞

⎠⎟

⎞⎞

⎠⎠ =

/

, and so forth.