Lesson 59: Using Fractional Exponents

New Concept

Exponents of the form $\frac{m}{n}$, where $m$ and $n$ are integers and $n \neq 0$, are called rational exponents.

What’s next

Next, you’ll apply this property to rewrite expressions, simplify radicals, and solve equations involving fractional exponents.

Write $5^{\frac{1}{4}}$ as a radical: $5^{\frac{1}{4}} = \sqrt[4]{5}$. The denominator 4 becomes the index of the root.
Write $8^{\frac{2}{3}}$ as a radical and simplify: $8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = (2)^2 = 4$. Take the cube root first, then square it.

Think of a fractional exponent as a two-part instruction! The bottom number (denominator) tells you which root to take, like a square root or cube root. The top number (numerator) tells you what power to raise the result to. It’s like telling your number, 'First get your roots, then get your power!' This makes tricky expressions manageable.

Radical expressions of the form $\sqrt[n]{a}$ are called nth roots. In this form n is the index and a is the radicand. $\sqrt[n]{a} = b$ if $b^n = a$. When nth roots have an odd index, there is only one real root. When they have an even index and a positive radicand, there are two real roots.

The cube root of 27: $\sqrt[3]{27} = 3$ because $3^3 = 27$. The index is odd, so there is one real root.
The fourth root of 81: $\sqrt[4]{81} = \pm 3$ because both $3^4 = 81$ and $(-3)^4 = 81$. The index is even.
The cube root of -8: $\sqrt[3]{-8} = -2$ because $(-2)^3 = -8$. The odd index allows a negative radicand.

The 'nth root' is like asking, 'What number, when multiplied by itself n times, gives me the number inside the radical?' The little 'n' is your guide. Be careful! An even index on a negative number is a no-go in the real number world because a real number multiplied by itself an even number of times can’t be negative.

Simplify $x^2 \cdot x^5$: Since the base is the same, add the exponents: $x^2 \cdot x^5 = x^{2+5} = x^7$.
Simplify $49^{\frac{1}{4}} \cdot 49^{\frac{3}{4}}$: $49^{\frac{1}{4}} \cdot 49^{\frac{3}{4}} = 49^{\frac{1}{4}+\frac{3}{4}} = 49^1 = 49$.

When you multiply two powers that share the same base, just keep the base and have a little party with the exponents by adding them together! This rule works for whole numbers and fractions alike. It’s a fantastic shortcut to combine expressions into one neat power, saving you from a lot of messy calculations and simplifying your work.