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Exponents

Understanding negative exponents

NEGATIVE EXPONENTS

$x^{-n}=\frac{1}{x^{n}}$    and    $\frac{1}{x^{-n}}=x^{n}$

When raising a base to a negative exponent, find the reciprocal of the base.

The definition indicates that negative exponents give us reciprocals, as the following examples illustrate.

Simplify 2-3

The first step is to rewrite the expression with a positive exponent, by using our definition. After that, we simplify.

 $2^{-3}=\frac{1}{2^{3}}$ Definition of negative exponents $=\frac{1}{8}$ The cube of 2 is 8

Simplify (-3)-2

The fact that the base in this problem is negative does not change the procedure we use to simplify:

 $(-3)^{-2}=\frac{1}{(-3)^{2}}$ Definition of negative exponents $=\frac{1}{9}$ The square of -3 is 9

Simplify $(\frac{1}{3})^{-1}$

$(\frac{1}{3})^{-1}=(\frac{3}{1})^{1}=\frac{3}{1}=3$

Simplify $-(\frac{2}{3})^{-2}$

$-(\frac{2}{3})^{-2}=-(\frac{3}{2})^{2}=-\frac{3\cdot3}{2\cdot2}=-\frac{9}{4}$