What Do Fractional Exponents Mean?
Fractions as exponents?! Don’t worry, it’s just notational shorthand for powers and roots. Once you understand it, they’re easy as pie! Check it out.
Recall this diagram?
The important feature here is the root index. Remember the root index tells us how many times our answer must be multiplied with itself to yield the radicand.
A fractional exponent is an alternate notation for expressing powers and roots together. For example, the following are equivalent.
We write the power in numerator and the index of the root in the denominator. If there is no power being applied, write “1” in the numerator as a placeholder.
Example
What would the following be equivalent to in radical notation?
For our purposes, it doesn’t matter if you write the second power on the 8 or on the cube root.
Because the cube root of 8 is 2, I prefer to take the root first and then apply the power.
Of course, the other order yields the same result.
Another Example
What is the following equivalent to in exponent notation?
This is equivalent to 2 raised to the 5/4’s power.
If we want to, we can manipulate the above expression even further. Begin by recognizing that 5/4 is equivalent to 1 + 1/4.
Using exponent properties from lesson twenty-nine, we can split it into two expressions involving base 2.
From there drop the power of 1 since it isn’t necessary and rewrite the 1/4 power as a root index of 4. Also feel free to drop the multiplication symbol.
There are many ways to write the same thing.
Many times I have had students come to me confused because they couldn’t figure out what they were doing wrong; their answer didn’t match the answer key. Often the answer key had simplified the answer further or wrote it in a different format than the student’s work. The student was correct but didn’t know it!
Thanks for reading! For more examples, including examples involving algebraic expressions, see the video linked above.
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