Lesson 2: Exponent Properties: Products, Quotients, and Powers

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

The multiplicative inverse (reciprocal) of $a^n$ is $a^{-n}$, and vice versa: $a^n \cdot a^{-n} = a^0 = 1$

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