Negative Indices and Power of a Power

When a power is raised to another power, multiply the indices: $(a^m{)}^n=a^{m\times n}$.

So $(2^3{)}^5=2^{3\times 5}=2^{15}$.

The indices are multiplied, not added, so $2^{3+5}=2^8$ is wrong. Both indices must be used, so $2^5$ ignores the power inside the bracket. Also, $2^3$ means $2\times 2\times 2=8$, not $2\times 3=6$, so the base stays as 2.

A negative index means take the reciprocal of the base: $a^{-1}=\frac{1}{a}$.

So $4^{-1}=\frac{1}{4}$.

The minus sign in the index does not make the answer negative, and it does not tell you to multiply the base by the index. It tells you to write 'one over' the base.

A negative index means take the reciprocal and make the index positive: $a^{-n}=\frac{1}{a^n}$.

So $2^{-3}=\frac{1}{2^3}=\frac{1}{8}$, because $2^3=2\times 2\times 2=8$.

The negative index means 'one over', so the answer is not $-8$, and it is not $8$ either. The minus sign cannot simply be ignored. Note also that $2^3$ is 8, not $2\times 3=6$.

A negative index means take the reciprocal and make the index positive: $a^{-n}=\frac{1}{a^n}$.

So $(-5{)}^{-3}=\frac{1}{(-5{)}^3}$. Since $(-5{)}^3=-5\times -5\times -5=-125$, the answer is $\frac{1}{-125}=-\frac{1}{125}$.

The answer is negative because the base is negative and the index 3 is odd, not because the index is negative. Dropping the minus on the base would wrongly give $\frac{1}{125}$, and forgetting the reciprocal would wrongly give $-125$.

When a power is raised to another power, multiply the indices: $(a^m{)}^n=a^{m\times n}$.

So $(5^{-2}{)}^2=5^{-2\times 2}=5^{-4}$, which equals $\frac{1}{625}$.

The indices are multiplied, not added. Adding would give $5^0=1$. Both indices must be used, so writing $\frac{1}{5^2}$ ignores the outer power of 2. A negative index makes a reciprocal, not a negative answer, so $-5^4$ is wrong.

A negative index on a fraction means flip the fraction and make the index positive: ${\left(\frac{a}{b}\right)}^{-n}={\left(\frac{b}{a}\right)}^n$.

So ${\left(\frac{1}{2}\right)}^{-3}={\left(\frac{2}{1}\right)}^3=2^3=8$.

Flipping the fraction first is what makes the answer bigger than 1. The negative index does not make the answer negative, and $2^3$ means $2\times 2\times 2=8$, not $2\times 3=6$.