   SEARCH HOME Math Central Quandaries & Queries  Question from Tori, a student: What is the square root of 729 to the third power? Hi Tori,

This is a fun problem. I think it has to do with punctuation. Your question is

What is the square root of 729 to the third power?

Is that the same as

What is the square root of 729, to the third power?

What is the square root of 7293?

What is √729 to the third power?

What is √(7293)?

and

What is (√729)3?

One more time, with fractional exponents

What is (7293)1/2?

and

What is (7291/2)3?

The laws of exponents tell us that these are the same question. They both ask

What is 7293/2?

The advantage of seeing the problem this way is that I have a choice of which way to make the calculation, either (7293)1/2 or (7291/2)3. The first looks daunting. I can multiply 729 729 729 but the product is large and taking its square root is not something I want to do. In the second expression I need to find the square root of 729. Using some experimentation, 202 = 400 so 20 is too small, 302 = 900 so 30 is too large, 252 = 625 so 25 is still to small. Thus the square root of 729 is between 25 and 30. With one or two more attempts I can find the square root of 729 and then all that is left is to cube the result.

I hope this helps,
Penny     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.