Lesson 1: Laws of Exponents

New Concept

Master three new laws of exponents for simplifying complex expressions. You'll learn how to handle powers of powers, products, and quotients, building on the rules you already know for multiplying and dividing powers with the same base.

What’s next

Next, you’ll see these laws in action with worked examples and apply them yourself in a series of interactive practice cards.

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 $2^3x^3$.

Property

To raise a quotient to a power, raise both the numerator and the denominator to the power. In symbols,

Examples

• To simplify $(\frac{a}{3})^4$, raise both the numerator and denominator to the fourth power: $\frac{a^4}{3^4} = \frac{a^4}{81}$.
• For $(\frac{x^2}{y^3})^2$, apply the outer exponent to both parts and simplify: $\frac{(x^2)^2}{(y^3)^2} = \frac{x^4}{y^6}$.
• Simplify $(\frac{-2}{z})^3$ by applying the exponent to the top and bottom: $\frac{(-2)^3}{z^3} = \frac{-8}{z^3}$.

Explanation

This is like distributing the exponent to a fraction. Raising a fraction to a power means multiplying the fraction by itself that many times. This results in the numerator and denominator each being raised to that power.

Property

To simplify complex expressions, combine the laws of exponents while following the order of operations. Always simplify powers before performing multiplication.

1. $a^m \cdot a^n = a^{m+n}$
2. $\frac{a^m}{a^n} = a^{m-n}$ or $\frac{1}{a^{n-m}}$
3. $(a^m)^n = a^{mn}$
4. $(ab)^n = a^nb^n$
5. $(\frac{a}{b})^n = \frac{a^n}{b^n}$

Examples

• Simplify $3a^2b(2ab^2)^3$. First, cube the term in parentheses: $3a^2b(8a^3b^6)$. Then multiply: $24a^{2+3}b^{1+6} = 24a^5b^7$.
• Simplify $(-y)^2(-yz)^3$. Simplify each power first: $y^2(-y^3z^3)$. Then multiply: $-y^{2+3}z^3 = -y^5z^3$.
• Simplify $(\frac{x^4}{2})^2(3x)^2$. Simplify powers: $(\frac{x^8}{4})(9x^2)$. Then multiply: $\frac{9x^{8+2}}{4} = \frac{9x^{10}}{4}$.

Explanation

When expressions have multiple operations, always follow the order of operations (PEMDAS). Simplify any powers first, such as $(3x^2)^3$, before you multiply that result by other terms in the expression.