Basic exponents and indices

An exponent records how many times a base is multiplied by itself, so $a^n$ is $a$ taken as a factor $n$ times. Working with exponents is mostly applying a handful of rules. When the bases match, multiplying adds the exponents: $a^m \cdot a^n = a^{m+n}$.

Dividing the same base subtracts the exponents, $\frac{a^m}{a^n} = a^{m-n}$, and raising a power to a power multiplies them, $(a^m)^n = a^{mn}$. A product inside a power splits across the factors: $(ab)^n = a^n b^n$.

Three special forms are worth memorising. Any nonzero base to the power $0$ is $1$, so $a^0 = 1$. A negative exponent is a reciprocal: $a^{-n} = \frac{1}{a^n}$, so $2^{-3} = \frac{1}{8}$. A fractional exponent is a root: $a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$, which is why $\sqrt{x} = x^{1/2}$ and $\frac{1}{x} = x^{-1}$.

These rules let you rewrite a messy expression as a clean power of one variable. To combine roots and powers, convert everything to fractional exponents and apply the rules above, using a common denominator for the exponents when they need to be added.

Simplify $\frac{p^3}{p^{-2}}$. Dividing the same base subtracts the exponents: $p^{3 - (-2)} = p^{3 + 2} = p^5$.

Rewrite $27^{-4/3}$. The negative exponent gives a reciprocal and the fraction gives a root: $27^{-4/3} = \frac{1}{27^{4/3}} = \frac{1}{(\sqrt[3]{27})^4} = \frac{1}{3^4} = \frac{1}{81}$.

Combine $\sqrt[5]{m^3} \cdot \sqrt[4]{m^3}$. Write the roots as fractional exponents, $m^{3/5} \cdot m^{3/4}$, then add over a common denominator of $20$: $\frac{3}{5} + \frac{3}{4} = \frac{12}{20} + \frac{15}{20} = \frac{27}{20}$, so the product is $m^{27/20}$.