Simplifying Expressions with Exponents
What Are Exponents?
An exponent tells you how many times to multiply a base by itself. In the expression $a^n$, the number $a$ is the base and $n$ is the exponent (or power).
$$a^n = \underbrace{a \times a \times a \times \cdots \times a}_{n \text{ times}}$$
📜 The Laws of Exponents
To simplify expressions with exponents, we use a set of powerful rules. Master these, and you can simplify almost anything!
$$a^m \cdot a^n = a^{m+n}$$
When multiplying powers with the same base, add the exponents.
Example: $x^3 \cdot x^5 = x^{3+5} = x^8$
$$\frac{a^m}{a^n} = a^{m-n}$$
When dividing powers with the same base, subtract the exponents.
Example: $\frac{y^7}{y^2} = y^{7-2} = y^5$
$$(a^m)^n = a^{m \cdot n}$$
When raising a power to another power, multiply the exponents.
Example: $(z^4)^3 = z^{4 \cdot 3} = z^{12}$
$$(ab)^n = a^n \cdot b^n$$
Distribute the exponent to each factor inside the parentheses.
Example: $(3x)^2 = 3^2 \cdot x^2 = 9x^2$
$$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$
Distribute the exponent to the numerator and denominator.
Example: $\left(\frac{x}{4}\right)^3 = \frac{x^3}{64}$
$$a^0 = 1 \quad (a \neq 0)$$
$$a^{-n} = \frac{1}{a^n}$$
Examples: $5^0 = 1$ and $x^{-3} = \frac{1}{x^3}$