Exponent rules

Japanese school year: Math II

What you learn

Learn how exponent rules extend from positive integers to negative numbers, fractions, and real exponents. These computational rules are fundamental for modeling rapid expansion, exponential decay, and scientific notation in science and engineering. Prior comfort with basic integer powers and polynomial operations will help you master these fractional exponents.

Go to practice

Key points

This rule states that when multiplying powers with the same base $a$ , you simply add the exponents $m$ and $n$ . For example, $2^3 \times 2^4 = 2^7$ .

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

This rule states that when raising a power $a^m$ to another power $n$ , you multiply the exponents together to obtain $a^{mn}$ . It simplifies repeated exponentiation.

(a^m)^n = a^{mn}

This defines a negative exponent $a^{-n}$ as the reciprocal of the positive power, $\frac{1}{a^n}$ (where $a \neq 0$ ). It allows expressions with denominators to be rewritten using exponents.

a^{-n} = \frac{1}{a^n}

This defines a fractional exponent $a^{\frac{m}{n}}$ as the $n$ -th root $\sqrt[n]{a^m}$ (where $a > 0$ ). It enables rewriting radicals in exponential form to apply standard exponent rules.

a^{\frac{m}{n}} = \sqrt[n]{a^m} \quad (a > 0)

Choose a set to practice.

Practice 1Coming soon