Lesson 3: Rational Exponents

New Concept

Explore rational exponents, where a fraction like $\frac{m}{n}$ in $a^{m/n}$ represents both a root and a power. You'll learn to evaluate these expressions, convert between exponential and radical forms, and use them to solve equations.

What’s next

Next, you'll tackle interactive examples for evaluating expressions and converting notations, followed by practice cards to master solving equations with rational exponents.

Property

For a positive base $a$ and a rational exponent $\frac{m}{n}$ where $n \neq 0$:

To compute $a^{m/n}$, you can either take the $n$th root of $a$ first and then raise it to the $m$th power, or raise $a$ to the $m$th power and then take the $n$th root. The denominator of the exponent is the root, and the numerator is the power.

Examples

• To evaluate $64^{2/3}$, we can take the cube root of 64 first, which is 4, and then square it: $(64^{1/3})^2 = 4^2 = 16$.

• For $-8^{4/3}$, the exponent applies only to the 8. We find $(8^{1/3})^4 = 2^4 = 16$, so the expression equals $-16$.

Property

A power with a fractional exponent can be written in radical form as follows:

This relationship allows conversion between exponential and radical notations. Typically, it is easier to convert from radical notation to fractional exponents to simplify expressions.

Examples

• To write $x^{3/5}$ in radical notation, the denominator 5 becomes the index of the root and the numerator 3 is the power: $\sqrt[5]{x^3}$.

• The expression $4b^{-2/3}$ is converted by first applying the negative exponent rule, then converting to radical form: $\frac{4}{b^{2/3}} = \frac{4}{\sqrt[3]{b^2}}$.

Property

Powers with rational exponents obey the same laws of exponents as powers with integer exponents. For a base $a > 0$ and rational exponents $p$ and $q$:

1. First Law (Product of Powers): $a^p \cdot a^q = a^{p+q}$
2. Second Law (Quotient of Powers): $\frac{a^p}{a^q} = a^{p-q}$
3. Third Law (Power of a Power): $(a^p)^q = a^{pq}$
4. Fourth Law (Power of a Product): $(ab)^p = a^p b^p$

Examples

• To simplify $x^{1/2} \cdot x^{1/4}$, we add the exponents: $x^{1/2 + 1/4} = x^{2/4 + 1/4} = x^{3/4}$.

Property

To solve an equation involving a variable raised to a rational exponent, first isolate the power. Then, raise both sides of the equation to the reciprocal of that exponent.

If $X^{m/n} = C$, then raise both sides to the power of $\frac{n}{m}$:

This process works because $(X^{m/n})^{n/m} = X^{(m/n) \cdot (n/m)} = X^1 = X$.

Examples

• To solve $x^{3/2} = 64$, raise both sides to the reciprocal power of $\frac{2}{3}$: $(x^{3/2})^{2/3} = 64^{2/3}$, which simplifies to $x = (\sqrt[3]{64})^2 = 4^2 = 16$.