Before you get started, take this readiness quiz.
Rational exponents are another way of writing expressions with radicals. When we use rational exponents, we can apply the properties of exponents to simplify expressions.
The Power Property for Exponents says that \((a^m)^n=a^\) when m and n are whole numbers. Let’s assume we are now not limited to whole numbers.
Suppose we want to find a number p such that \((8^p)^3=8\). We will use the Power Property of Exponents to find the value of p.
But we know also \((\sqrt[3])^3=8\). Then it must be that \(8^>=\sqrt[3]\)
This same logic can be used for any positive integer exponent n to show that \(a^>=\sqrt[n]\).
There will be times when working with expressions will be easier if you use rational exponents and times when it will be easier if you use radicals. In the first few examples, you’ll practice converting expressions between these two notations.
Write as a radical expression:
We want to write each expression in the form \(\sqrt[n]\).
| 1. | \(x^>\) |
| The denominator of the exponent is 2, so the index of the radical is 2. We do not show the index when it is 2. | \(\sqrt\) |
| 2. | \(y^>\) |
| The denominator of the exponent is 3, so the index is 3. | \(\sqrt[3]\) |
| 3. | \(z^\frac>\) |
| The denominator of the exponent is 4, sothe index is 4. | \(\sqrt[4]\) |
Write as a radical expression:
