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Fractional Exponents, a la Shmoop. When inspected under a microscope…

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…fractional exponents are just like other exponents…

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…except they’re smaller, they’re meaner…

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… and they can spread throughout a math quiz like germs on a chew-toy.

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Fractional exponents are abbreviations for taking the roots of a number.

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For example, x to the power of one-sixth means the sixth root of x.

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But what about those fractions that have a number other than 1 in the numerator?

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What would 8 to the power of two-thirds mean?

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According to the Rules of Multiplying Exponents:

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Let’s say we raise x to the two-thirds power by 3.

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That means we multiply x to the power of two-thirds by itself 3 times...

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…adding 2/3 plus 2/3 plus 2/3 gives us 6/3. To simplify it, we divide 6 by 3 to get 2...

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…which means our x is squared.

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Therefore, x to the power of two-thirds is still a root.

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It's the cubed root of x squared. Which makes 8 to the power of two-thirds the

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cubed root of 8 squared.

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Since the cube root of 8 is 2, or 2 times 2 times 2, this simplifies to two to the power

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of 2

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Which equals 4. So, as you can see, fractional exponents really

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aren’t that scary.

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Certainly less painful than a

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root canal.

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