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?
![](https://stayfreemagazine.org/wp-content/uploads/2021/07/what-do-fractional-exponents-mean.png)
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.
![](https://stayfreemagazine.org/wp-content/uploads/2021/07/what-do-fractional-exponents-mean-1.png)
Example
What would the following be equivalent to in radical notation?
![](https://stayfreemagazine.org/wp-content/uploads/2021/07/what-do-fractional-exponents-mean-2.png)
For our purposes, it doesn?t matter if you write the second power on the 8 or on the cube root.
![](https://stayfreemagazine.org/wp-content/uploads/2021/07/what-do-fractional-exponents-mean-3.png)
Because the cube root of 8 is 2, I prefer to take the root first and then apply the power.
![](https://stayfreemagazine.org/wp-content/uploads/2021/07/what-do-fractional-exponents-mean-4.png)
Of course, the other order yields the same result.
![](https://stayfreemagazine.org/wp-content/uploads/2021/07/what-do-fractional-exponents-mean-5.png)
Another Example
What is the following equivalent to in exponent notation?
![](https://stayfreemagazine.org/wp-content/uploads/2021/07/what-do-fractional-exponents-mean-6.png)
This is equivalent to 2 raised to the 5/4?s power.
![](https://stayfreemagazine.org/wp-content/uploads/2021/07/what-do-fractional-exponents-mean-7.png)
If we want to, we can manipulate the above expression even further. Begin by recognizing that 5/4 is equivalent to 1 + 1/4.
![](https://stayfreemagazine.org/wp-content/uploads/2021/07/what-do-fractional-exponents-mean-8.png)
Using exponent properties from lesson twenty-nine, we can split it into two expressions involving base 2.
![](https://stayfreemagazine.org/wp-content/uploads/2021/07/what-do-fractional-exponents-mean-9.png)
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.
![](https://stayfreemagazine.org/wp-content/uploads/2021/07/what-do-fractional-exponents-mean-10.png)
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!
![](https://stayfreemagazine.org/wp-content/uploads/2021/07/what-do-fractional-exponents-mean-11.png)
Thanks for reading! For more examples, including examples involving algebraic expressions, see the video linked above.
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