Rational exponents are the same thing as fractional exponents. You can use either word, but “rational” is the official term.
On my fractional exponents page, I explained how to simplify basic fractional exponents with a 1 in the numerator. If you haven’t read that page yet, I recommend reading it before reading this page.
This page is all about how to simplify more advanced rational exponents that have something other than a 1 in the numerator.
\[x^\frac{a}{b}=\sqrt[b]{x^{a}}\]
OR
\[x^\frac{a}{b}=(\sqrt[b]{x})^{a}\]
When you have a power with a rational exponent, the denominator of the fraction represents the type of root. And the base of the power is the argument of the root.
The numerator of the fraction works like a normal exponent. You can raise the entire root to the numerator exponent OR you can put the exponent inside the root as part of the argument.
Both are correct, so you can choose the option that looks easiest to you.
Examples
\(8^\frac{2}{3}=\sqrt[3]{8^{2}}\)
OR
\(8^\frac{2}{3}=(\sqrt[3]{8})^{2}\)
\(16^\frac{5}{2}=\sqrt{16^{5}}\)
OR
\(16^\frac{5}{2}=(\sqrt{16})^{5}\)
\(x^\frac{4}{7}=\sqrt[7]{x^{4}}\)
OR
\(x^\frac{4}{7}=(\sqrt[7]{x})^{4}\)
I usually put the exponent outside of the root when I’m working with numbers because it makes the roots easier to evaluate.
When I’m working with variables, I like to keep the exponent inside the root because it looks more neat and clean.
Ultimately, you’ll get the same answer either way so it doesn’t matter which one you choose. Just choose the easiest option for the problem you’re working on.
How to Simplify Rational Exponents
- Use the denominator of the rational exponent to identify the type of root (square root, cube root, 4th root etc.).
- Write the base of the power inside the root.
- Write the numerator exponent inside OR outside the root.
- Evaluate the expression, if possible.
Simplify \(625^{\frac{3}{4}}\)
The denominator of the rational exponent is 4 and the base of the power is 625. So, I’ll write 625 inside of a 4th root.
\(\sqrt[4]{625}\)
The numerator of the rational exponent is 3.
I can write it inside the root as \(\sqrt[4]{625^3}\)
OR
I can write it outside the root as \((\sqrt[4]{625})^3\).
In my opinion, the second option looks easiest to evaluate, so I’ll write the exponent outside the root.
\(625^{\frac{3}{4}}= (\sqrt[4]{625})^3\)
Next, I’ll evaluate the root.
5 times itself four times is 625, so the fourth root of 625 is 5.
\(\sqrt[4]{625}=5\)
Then I’ll raise 5 to the third power.
\((5)^{3} = 5\times 5\times 5 = 125\)
Answer: \(625^{\frac{3}{4}}=125\)
If I’d chosen the first option, I would get the same answer because…
\(625^3 = 244140625\)
And the fourth root of that number is 125.
\(\sqrt[4]{244140625}=125\)
Why can you write the numerator exponent inside or outside the root?
The reverse power rule of exponents is the reason the numerator exponent can be written inside or outside of the root.
Reverse Power Rule of Exponents
\[x^{ab}=(x^{a})^{b}\]
If you have a power like \(y^{\frac{2}{3}}\), you can think of the exponent as…
\(2(\frac{1}{3})\)
\(\frac{1}{3}(2)\)
The reverse power rule of exponents allows us to write the rational exponent as a power raised to another power, which gives us…
\((y^{2})^{\frac{1}{3}} = \sqrt[3]{y^{2}}\)
\((y^{\frac{1}{3}})^{2}=(\sqrt[3]{y})^{2}\)
That’s why you can write the numerator exponent inside or outside of the root.
How to Write Radicals in Exponential Form
It can be helpful to write radicals as powers with rational exponents. To do this…
- Identify the argument of the root and write it as the base of the power.
- Identify the type of root and write it as the denominator of the exponent.
- Simplify the expression as much as possible using the exponent rules.
Write \(\sqrt[4]{x^3}\) in exponential form.
The argument of the root is \(x^{3}\), so that will be the base of the power.
The root is a 4th root, so 4 is the denominator of the exponent.
\(\sqrt[4]{x^3}=(x^{3})^{\frac{1}{4}}\)
The power rule of exponents says that when we have a power raised to an exponent, we can multiply the two exponents.
\((x^{3})^{\frac{1}{4}}=x^{\frac{3}{4}}\)
Answer: \(\sqrt[4]{x^3}=x^{\frac{3}{4}}\)
Multiple Exponents or Bases
If you have multiple exponents or bases, you can use the exponent rules to simplify the overall expression before you simplify the fractional exponent.
Just remember to follow the rules for adding, subtracting, multiplying and dividing fractions when you apply the exponent rules.
