Exponent rules tutorial
Exponents let you write repeated multiplication in a compact way. Mastering the rules for multiplying, dividing, and raising powers to powers is essential for algebra and beyond. This tutorial covers the product, quotient, and power rules with worked examples and tips for avoiding common mistakes.
Core rules
Product rule: $a^m \times a^n = a^{m+n}$
Quotient rule: $a^m \div a^n = a^{m-n}$
Power rule: $(a^m)^n = a^{mn}$
What is an exponent?
An exponent tells you how many times to multiply a number by itself. For example, $2^4 = 2 \times 2 \times 2 \times 2 = 16$.
Exponents appear in algebra, science, and computer science. Understanding their rules makes simplifying expressions and solving equations much easier.
Why do the rules work?
Each rule comes from expanding the repeated multiplication. For example, $a^3 \times a^2 = (a \times a \times a) \times (a \times a) = a^5$.
Worked examples
- Product rule: $x^3 \times x^5 = x^{3+5} = x^8$
- Quotient rule: $y^7 \div y^2 = y^{7-2} = y^5$
- Power rule: $(z^2)^4 = z^{2\times4} = z^8$
Tips and common mistakes
- Only combine exponents when the base is the same. $x^3 \times y^3$ cannot be combined.
- When dividing, subtract the exponents: $a^m \div a^n = a^{m-n}$.
- When raising a power to a power, multiply the exponents: $(a^m)^n = a^{mn}$.
- Negative exponents mean reciprocals: $a^{-n} = 1/a^n$.
- Any nonzero number to the zero power is 1: $a^0 = 1$ (for $a \neq 0$).
- Zero to any positive power is zero: $0^n = 0$ for $n > 0$.
Exponent rules in real life
Exponents are used in science (scientific notation, exponential growth), finance (compound interest), and computer science (algorithm complexity, data storage). Mastering the rules makes it easier to work with large and small numbers, simplify formulas, and solve equations.
- Scientific notation: $3.2 \times 10^5$
- Compound interest: $A = P(1 + r/n)^{nt}$
- Algorithm complexity: $O(2^n)$
Try it yourself
Test your understanding with a practice question. Enter your answer as an exponent (e.g., x^5).