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Negative Exponent Property

Suppose \(\large{a}\) and \(\large{b}\) are nonzero real numbers and \(\large{n}\) is a positive integer,

$${a^{ - n}} = {1 \over {{a^n}}}$$

$${1 \over {{a^{ - n}}}} = {a^n}$$

$${\left( {{a \over b}} \right)^{ - n}} = {\left( {{b \over a}} \right)^n}$$

Description: When a nonzero base \(a\) is raised to a negative power, take the reciprocal of the base then switch the sign of the exponent from negative to positive.

Some examples:

$${3^{ - 2}} = {1 \over {{3^2}}}$$

$${1 \over {{7^{ - 5}}}} = {7^5}$$

$${\left( {{2 \over 5}} \right)^{ - 3}} = {\left( {{5 \over 2}} \right)^3}$$

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Laws of Exponents
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