Laws of Exponents

Exponents are shorthand for repeated multiplication. Instead of writing $2 \times 2 \times 2 \times 2$, you can write $2^4$. The small raised number (4) is the exponent, and it tells you how many times to multiply the base (2) by itself.

Working with exponents means following a small set of rules. These rules aren't arbitrary — they come directly from what exponents mean, and once you've seen the reasoning, they're easy to remember.

Basic Exponent Notation

$$a^n = a \times a \times a \times ... \times a \text{ (n times)}$$

Example: $3^4 = 3 \times 3 \times 3 \times 3 = 81$

Example: $x^5 = x \times x \times x \times x \times x$

The base can be any number or variable. The exponent must be a whole number for now; zero and negative exponents come later in this lesson.

Product Rule: Multiplying Powers with the Same Base

When you multiply two powers that have the same base, you add the exponents.

$$a^m \times a^n = a^{m+n}$$

Why? Because $a^m$ means $a$ multiplied by itself $m$ times, and $a^n$ means $a$ multiplied by itself $n$ times. When you multiply them together, you're multiplying $a$ by itself $m + n$ times total.

Example: $x^3 \times x^4 = x^{3+4} = x^7$

Think about it: $x^3 = x \times x \times x$ and $x^4 = x \times x \times x \times x$. Together that's $x$ multiplied by itself 7 times.

Example: $2^5 \times 2^3 = 2^{5+3} = 2^8 = 256$

Example: $y^2 \times y = y^2 \times y^1 = y^{2+1} = y^3$

Notice that $y$ by itself is the same as $y^1$.

Quick check: what does $a^4 \times a^3$ simplify to? Show answer$a^{4+3} = a^7$. Just add the exponents — the base stays the same.

Quotient Rule: Dividing Powers with the Same Base

When you divide two powers with the same base, you subtract the exponents.

$$\frac{a^m}{a^n} = a^{m-n}$$

Example: $\frac{x^7}{x^3} = x^{7-3} = x^4$

This makes sense: $\frac{x \times x \times x \times x \times x \times x \times x}{x \times x \times x}$. Three $x$'s cancel from top and bottom, leaving $x^4$.

Example: $\frac{5^6}{5^2} = 5^{6-2} = 5^4 = 625$

Example: $\frac{a^8}{a^5} = a^{8-5} = a^3$

Power Rule: Raising a Power to a Power

When you raise a power to another power, you multiply the exponents.

$$(a^m)^n = a^{m \times n}$$

Example: $(x^2)^3 = x^{2 \times 3} = x^6$

Why? $(x^2)^3$ means $x^2 \times x^2 \times x^2$. Using the product rule, that's $x^{2+2+2} = x^6$.

Example: $(3^2)^4 = 3^{2 \times 4} = 3^8 = 6561$

Example: $(y^5)^2 = y^{10}$

Power of a Product

When you raise a product to a power, you raise each factor to that power.

$$(ab)^n = a^n b^n$$

Example: $(2x)^3 = 2^3 x^3 = 8x^3$

Example: $(3y)^4 = 3^4 y^4 = 81y^4$

Example: $(xy)^5 = x^5 y^5$

Power of a Quotient

When you raise a quotient to a power, you raise both numerator and denominator to that power.

$$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$

Example: $\left(\frac{x}{2}\right)^3 = \frac{x^3}{2^3} = \frac{x^3}{8}$

Example: $\left(\frac{3}{y}\right)^2 = \frac{3^2}{y^2} = \frac{9}{y^2}$

The Zero Exponent

Any non-zero number raised to the zero power equals 1.

$$a^0 = 1 \quad (a \neq 0)$$

This might seem strange, but it follows from the quotient rule. Consider $\frac{a^3}{a^3}$. On one hand, anything divided by itself is 1. On the other hand, using the quotient rule: $\frac{a^3}{a^3} = a^{3-3} = a^0$. So $a^0 = 1$.

Example: $5^0 = 1$

Example: $x^0 = 1$ (as long as $x \neq 0$)

Example: $(3xy^2)^0 = 1$

Negative Exponents

A negative exponent means "reciprocal."

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

Example: $x^{-3} = \frac{1}{x^3}$

Example: $2^{-4} = \frac{1}{2^4} = \frac{1}{16}$

Example: $\frac{1}{y^{-2}} = y^2$

To move a factor from numerator to denominator (or vice versa), change the sign of the exponent.

Combining Multiple Rules

Most problems require using several exponent rules together.

Example: Simplify $(x^2 y^3)^2$.

Apply power of a product: $$= x^4 y^6$$

Example: Simplify $\frac{(2a^3)^2}{a^4}$.

Power of a product on top: $$= \frac{4a^6}{a^4}$$

Quotient rule: $$= 4a^{6-4} = 4a^2$$

Example: Simplify $x^{-2} \times x^5$.

Product rule: $$= x^{-2+5} = x^3$$

Example: Simplify $\frac{15x^7 y^2}{3x^3 y^5}$.

Divide coefficients: $$= 5 \times \frac{x^7}{x^3} \times \frac{y^2}{y^5}$$

Apply quotient rule: $$= 5x^4 y^{-3} = \frac{5x^4}{y^3}$$

Put It Into Practice

Simplify each expression:

$x^5 \times x^3$ Show answer$x^{5+3} = x^8$ (product rule — add exponents)
$\frac{y^9}{y^4}$ Show answer$y^{9-4} = y^5$ (quotient rule — subtract exponents)
$(a^3)^4$ Show answer$a^{3 \times 4} = a^{12}$ (power rule — multiply exponents)
$(2m)^3$ Show answer$2^3 m^3 = 8m^3$ (power of a product — apply exponent to each factor)
$n^0$ Show answer$1$ — any non-zero base raised to the zero power equals 1
$x^{-5}$ Show answer$\frac{1}{x^5}$ — negative exponent means reciprocal
$\frac{10a^8 b^3}{5a^2 b^7}$ Show answerDivide the coefficients: $\frac{10}{5} = 2$. Apply quotient rule: $a^{8-2} = a^6$ and $b^{3-7} = b^{-4}$. Result: $\frac{2a^6}{b^4}$
$(x^2 y^{-3})^2$ Show answerApply the exponent to each factor: $x^{2 \times 2} y^{-3 \times 2} = x^4 y^{-6} = \frac{x^4}{y^6}$

What's Next?

A handful of patterns trip students up most often. The product rule applies only when the bases match: $x^2 \times x^3 = x^5$ (add the exponents), but $x^2 \times y^3$ can't be combined at all because $x$ and $y$ are different bases. When you raise a product to a power, the exponent reaches every factor: $(3x)^2 = 9x^2$, not $3x^2$. A zero exponent gives 1, not 0 — $5^0 = 1$, $x^0 = 1$, $(374)^0 = 1$. And a negative exponent means reciprocal, not "negative answer": $x^{-3} = \frac{1}{x^3}$, which is positive whenever $x$ is.

The rules here carry forward into scientific notation, polynomials, and eventually into exponential and logarithmic functions. The arithmetic stays the same; only the contexts change.