Understanding Exponent Properties

  • by Brett Berry - Thu, 12/01/2016 - 12:59

An Unusual Number Problem

Wilfred heard that there is only one number between 2 and 200,000,000,000,000 which is a perfect square, a perfect cube, and a fifth power, and has decided to look for it.

So far he has checked every number up to 100,000 and is beginning to get somewhat discouraged. Perhaps you can help him find it?

— This puzzle came from “Puzzles in Math & Logic” by Aaron J. Friedland

Need Help Getting Started?
If you’re just joining us, Welcome! I introduced exponents in lesson two. Check it out if you need a refresher!

Today we’re covering properties of exponents (the key to solving the above problem!)

Exponent Properties
Recall an exponent describes the number of times a value is multiplied with itself. For example, six x’s multiplied together is the same as “x to the sixth power.”


Using the associative property, regroup the terms.


(x • x) equals x-squared, so the expression is the same as having three x-squared’s multiplied together.


This is the same as “x-squared cubed” or “x-squared raised to the third power.”


This demonstrates two properties of exponents:
To multiply terms with exponents together make sure the bases are all the same, in this case they were all x, then add the exponents (see diagram 3).
When raising a term with an exponent to another exponent, multiply the powers together. For example:

I could also work backwards.
Start with x raised to the sixth power. Since 6 = 3 x 2, rewrite the expression like this:

X-cubed raised to the second power means I am multiplying two x-cubes together.

Which is equal to six x’s multiplied together.

That should get you started in the right direction!

Brett Berry is a math evangelist who writes a math blog for Medium.com called Math Memoirs.