Laws of Exponents

The three core rules

• Same base, multiplying: add the exponents: $x^m \cdot x^n = x^{m+n}$. E.g. $2^4 \cdot 2^3 = 2^{7} = 128$.
• A power raised to a power: multiply the exponents: $(x^m)^n = x^{mn}$. E.g. $(3^2)^5 = 3^{10} = 59049$.
• Negative exponent: take the reciprocal: $x^{-m} = \dfrac{1}{x^m}$. E.g. $2^{-5} = \dfrac{1}{32}$.

Worked example: a negative exponent on a fraction

Simplify $\left(\dfrac{-3}{4}\right)^{-3}$.

Solution: A negative exponent on a fraction flips it, then applies the (now positive) power:

Worked example: combining rules to avoid big numbers

Simplify $\dfrac{7^8 \times 12^8}{14^8 \times 6^8}$ without computing any of these to the 8th power directly.

Solution: Notice $7 \times 12 = 84 = 14 \times 6$, the numerator and denominator are built from the same product, just split differently. Since $a^n b^n = (ab)^n$:

Worked example: solving for an exponent

If $\left(\dfrac{5}{3}\right)^{-5} \times \left(\dfrac{5}{3}\right)^{11} = \left(\dfrac{5}{3}\right)^{8x}$, find $x$.

Solution: Same base on both sides, so just add the exponents on the left and match: