Negative Exponents and Power of a Power

When raising a number to a power and then raising it again, multiply the exponents. The formula is $(a^m{)}^n=a^{m\times n}$. So, $(2^3{)}^5=2^{3\times 5}=2^{15}$.

A negative exponent means you take the reciprocal of the base and change the exponent to positive. The formula is $a^{-1}=\frac{1}{a}$. So, $4^{-1}=\frac{1}{4}$.

A negative exponent means you take the reciprocal of the base and change the exponent to positive. The formula is $a^{-n}=\frac{1}{a^n}$. So, $2^{-3}=\frac{1}{2^3}=\frac{1}{8}$.

A negative exponent means you flip the base and make the exponent positive. The formula is $a^{-n}=\frac{1}{a^n}$. So, $(-5{)}^{-3}=\frac{1}{(-5{)}^3}=-\frac{1}{125}$.

When raising a number with a negative exponent to another power, multiply the exponents. The formula is $(a^m{)}^n=a^{m\times n}$. So, $(5^{-2}{)}^2=5^{-2\times 2}=5^{-4}$.

A negative exponent means you flip the base and make the exponent positive. The formula is $(\frac{a}{b}{)}^{-n}=(\frac{b}{a}{)}^n$. So, $(\frac{1}{2}{)}^{-3}=2^3=8$.