Properties of Integer Exponents

Exponents are a shorthand way to show repeated multiplication. When working with expressions that have exponents, there are several key rules—or properties—that make simplifying them much easier.

The Product Rule

When you multiply two powers that have the same base, you add their exponents. $a^m \cdot a^n = a^{m+n}$

Example: Simplify $x^3 \cdot x^5$. Since the bases are both $x$, we just add the exponents: $x^3 \cdot x^5 = x^{3+5} = x^8$

The Quotient Rule

When you divide two powers with the same base, you subtract the exponent in the denominator from the exponent in the numerator. $\frac{a^m}{a^n} = a^{m-n}$

Example: Simplify $\frac{y^7}{y^2}$. $\frac{y^7}{y^2} = y^{7-2} = y^5$

Power of a Power Rule

When you raise a power to another power, you multiply the exponents. $(a^m)^n = a^{m \cdot n}$

Example: Simplify $(2^4)^3$. $(2^4)^3 = 2^{4 \cdot 3} = 2^{12}$

Power of a Product Rule

When you raise a product (terms being multiplied) to a power, the exponent applies to every factor inside the parentheses. $(ab)^m = a^m b^m$

Example: Simplify $(3x)^2$. $(3x)^2 = 3^2 \cdot x^2 = 9x^2$

Zero and Negative Exponents

Since we are dealing with integer exponents, the exponents can be zero or negative:

• Zero Exponent Rule: Any non-zero base raised to the power of $0$ is exactly $1$. $a^0 = 1$
• Negative Exponent Rule: A negative exponent means taking the reciprocal of the base. $a^{-n} = \frac{1}{a^n}$