7-1 Properties of Exponents

Property

A monomial is a number, a variable, or the product of a number and one or more variables. Monomials act as the individual terms or building blocks that make up larger algebraic expressions.

Examples

Property

When multiplying powers with the same base, add the exponents: $a^m \cdot a^n = a^{m+n}$.
When dividing powers with the same base, subtract the exponent of the denominator from the exponent of the numerator: $\frac{a^m}{a^n} = a^{m-n}$.

Examples

• Product Property: Multiply $10^4 \cdot 10^2$.

$10^4 \cdot 10^2 = 10^{4+2} = 10^6$.

• Quotient Property: Divide $\frac{10^7}{10^3}$.

$\frac{10^7}{10^3} = 10^{7-3} = 10^4$.

• Using Negative Exponents: Multiply $x^{-5} \cdot x^2$.

$x^{-5} \cdot x^2 = x^{-5+2} = x^{-3}$, which can be written as $\frac{1}{x^3}$.

Explanation

These exponent properties act as mathematical shortcuts because multiplication is just repeated addition, and division is repeated subtraction. When you multiply powers of the same base, you are combining groups of factors, so you add the exponents. When you divide, you are canceling out groups of factors, so you subtract the exponents.

Property

To raise a product to a power, raise each factor to the power. In symbols,

Examples

• To simplify $(4xy)^2$, apply the exponent to each factor inside: $4^2x^2y^2 = 16x^2y^2$.
• For $(-3a^2)^3$, raise each factor to the third power: $(-3)^3(a^2)^3 = -27a^6$.
• Note the difference: in $5x^3$, only $x$ is cubed. In $(5x)^3$, both 5 and $x$ are cubed, giving $125x^3$.

Explanation

This rule works because multiplication is commutative. An expression like $(2x)^3$ means $(2x)(2x)(2x)$. You can regroup the factors as $(2 \cdot 2 \cdot 2)(x \cdot x \cdot x)$, which is simply $2^3x^3$.