Product Rule

If multiplying two powers with the same base, add the exponents.

a^{m} \cdot a^{n} = a^{m + n}

Example:
$$
x^3 \cdot x^5 = x^{3+5} = x^8
$$

Quotient Rule

If dividing two powers with the same base, subtract the exponents.

\frac{a ^{m}}{a ^{n}} = a^{m - n}, a \neq = 0

Example:
$$
\frac{y^7}{y^4} = y^{7-4} = y^3
$$

If raising a power to another power, multiply the exponents.

(a^{m})^{n} = a^{m \cdot n}

Example:
$$
(x^2)^3 = x^{2 \cdot 3} = x^6
$$

Any nonzero number raised to the power of zero is 1.

a^{0} = 1, a \neq = 0

Example:
$$
5^0 = 1
$$

A negative exponent means the reciprocal of the base with a positive exponent.

a^{- n} = \frac{1}{a ^{n}}, a \neq = 0

Example:
$$
2^{-3} = \frac{1}{2^3} = \frac{1}{8}
$$

A fractional exponent represents a root.

a^{\frac{m}{n}} = n a^{m}

Example:
$$
8^{\frac{1}{3}} = \sqrt[3]{8} = 2
$$