Lesson 2: Exponent Rules

Property

To raise a power to another power, you multiply the exponents. The general form is $(a^m)^n = a^{m \cdot n}$.

Examples

• $(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$
• $(10^5)^2 = 10^{5 \cdot 2} = 10^{10}$
• $(x^6)^3 = x^{6 \cdot 3} = x^{18}$

Explanation

The power of a power rule simplifies expressions where an exponential term is raised to another exponent. For example, $(2^3)^4$ means you are multiplying $2^3$ by itself four times: $2^3 \cdot 2^3 \cdot 2^3 \cdot 2^3$. Using the product rule, you would add the exponents: $2^{3+3+3+3} = 2^{12}$. This rule provides a shortcut by simply multiplying the two exponents, $3 \cdot 4$, to get the same result.

Property

For any real number $a$ and natural numbers $m$ and $n$, the product rule of exponents states that

Examples

• To simplify $g^4 \cdot g^2$, we add the exponents: $g^{4+2} = g^6$.
• To simplify $(-5)^4 \cdot (-5)$, we recognize that $(-5)$ is $(-5)^1$. So, we have $(-5)^{4+1} = (-5)^5$.
• We can combine multiple terms: $y^3 \cdot y^6 \cdot y^2 = y^{3+6+2} = y^{11}$.

Explanation

When multiplying terms with the same base, keep the base and add the exponents. This is a shortcut for counting all the individual factors. For example, $x^2 \cdot x^3$ means $(x \cdot x) \cdot (x \cdot x \cdot x)$, which is simply $x^5$.

Property

To divide two powers with the same base, subtract the exponent of the denominator from the exponent of the numerator. For any non-zero number $a$, and for whole numbers $m$ and $n$ where $m > n$:

Examples