Section 8-3

Rational Exponents

POWERS OF 1/2

You know from the properties of exponents that:

a^b · a^c = a^{b + c}

You can use this to show that 9^1/2 = ±3:

9^1/2 · 9^1/2 = 9^{1/2 + 1/2} = 9¹

So,

9^1/2 · 9^1/2 = 9

What number multiplied by itself equals 9? There are two answers: 3 and –3.

3 · 3 = 9

(–3) · (–3) = 9

So, raising a number to the power of 1/2 works almost the same as square roots (except that, with square roots, by convention, we usually only mean the positive answer).

OTHER FRACTIONAL POWERS

You can use the same property of exponents to show that 8^1/3 = 2:

8^1/3 · 8^1/3 · 8^1/3 = 8^{1/3 + 1/3 + 1/3} = 8¹

So,

8^1/3 · 8^1/3 · 8^1/3 = 8

What number multiplied by itself three times equals 8? The only answer in this case is 2.

2 · 2 · 2 = 8

By a similar logic, 8^2/3 = 4, 8^4/3 = 16, 8^5/3 = 32, etc. (Convince yourself of this!)

CUBE ROOTS AND OTHER RADICALS

Fractional exponents can also be written as radicals:

The only difference is that, in most books, a radical expression is always positive, whereas the rational exponent may yield a positive or negative answer.

It's a good idea to review the properties of exponents before working through the problems in this section.