Laws of Exponents

What is an Exponent?

An exponent is shorthand for repeated multiplication.

$x^3 = x \times x \times x$

The exponent tells you how many times to multiply the base by itself.

The Six Laws

Product Rule

Same base? Add the exponents.

$x^a \cdot x^b = x^{a+b}$

Why it works: Count the factors.

$x^2 \cdot x^3 = \underbrace{(x \cdot x)}_{2} \cdot \underbrace{(x \cdot x \cdot x)}_{3} = x^5$

Examples:

• $x^4 \cdot x^2 = x^6$
• $y \cdot y^4 = y^5$ (remember: $y = y^1$)
• $2^3 \cdot 2^5 = 2^8 = 256$

Quotient Rule

Same base? Subtract the exponents.

$\frac{x^a}{x^b} = x^{a-b}$

Why it works: Cancel the factors.

$\frac{x^5}{x^2} = \frac{x \cdot x \cdot x \cdot x \cdot x}{x \cdot x} = x \cdot x \cdot x = x^3$

Examples:

• $\dfrac{x^7}{x^3} = x^4$
• $\dfrac{2^{10}}{2^6} = 2^4 = 16$

Power Rule

Power of a power? Multiply the exponents.

$(x^a)^b = x^{ab}$

Why it works: You’re repeating the multiplication.

$(x^2)^3 = x^2 \cdot x^2 \cdot x^2 = x^{2+2+2} = x^6$

Examples:

• $(x^3)^4 = x^{12}$
• $(2^2)^5 = 2^{10} = 1024$

Zero Exponent

Anything to the power of 0 is 1.

$x^0 = 1 \quad \text{(for } x \neq 0 \text{)}$

Why it works: Use the quotient rule.

$\frac{x^3}{x^3} = x^{3-3} = x^0$

But also $\dfrac{x^3}{x^3} = 1$, so $x^0 = 1$.

Negative Exponent

Negative exponent means reciprocal.

$x^{-a} = \frac{1}{x^a}$

Why it works: Use the quotient rule.

$\frac{x^2}{x^5} = x^{2-5} = x^{-3}$

But also:

$\frac{x^2}{x^5} = \frac{1}{x^3}$

So $x^{-3} = \dfrac{1}{x^3}$.

Examples:

• $2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8}$
• $x^{-1} = \dfrac{1}{x}$

Distributive Over Products

Power of a product? Distribute to each factor.

$(xy)^a = x^a \cdot y^a$

Why it works:

Examples:

• $(2x)^3 = 2^3 \cdot x^3 = 8x^3$
• $(3y^2)^2 = 9y^4$
• $\left(\dfrac{x}{y}\right)^2 = \dfrac{x^2}{y^2}$