Number - Powers and Roots

A power (or index) tells us how many times a base number is multiplied by itself. For example, in $a^n$, $a$ is the base and $n$ is the index.

Square numbers are the result of an integer multiplied by itself, such as $1, 4, 9, 16, 25, \dots$. The square root $\sqrt{x}$ is the inverse operation.

Cube numbers are the result of an integer multiplied by itself three times, such as $1, 8, 27, 64, 125, \dots$. The cube root $\sqrt[3]{x}$ is the inverse operation.

The Multiplication Law: When multiplying terms with the same base, we add the indices: $a^m \times a^n = a^{m+n}$.

The Division Law: When dividing terms with the same base, we subtract the indices: $a^m \div a^n = a^{m-n}$.

Power of a Power: When a power is raised to another power, we multiply the indices: $(a^m)^n = a^{mn}$.

Zero Index: Any non-zero base raised to the power of zero is equal to $1$.

Negative Indices: A negative index indicates a reciprocal: $a^{-n} = \frac{1}{a^n}$.

Fractional Indices: The denominator of a fractional index indicates the root, while the numerator indicates the power: $a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$.