Lesson 1-3: Powers of Monomials

Property

To raise a power to a power, keep the same base and multiply the exponents. In symbols,

Examples

• To simplify $(x^3)^5$, you multiply the exponents: $x^{3 \cdot 5} = x^{15}$.
• To simplify $(4^2)^3$, you keep the base and multiply the powers: $4^{2 \cdot 3} = 4^6$.
• Be careful to distinguish from products: $(a^5)(a^2) = a^{5+2} = a^7$, but $(a^5)^2 = a^{5 \cdot 2} = a^{10}$.

Explanation

Think of this as repeated multiplication. $(x^4)^3$ is just $x^4$ multiplied by itself three times. Adding the exponents $4+4+4$ is the same as multiplying $4 \cdot 3$. So, you multiply 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 $2^3x^3$.