Find the root

Question

Find the root.

⁴√1296

Answer 1

Hi Theresa,

 

Hope this helps you figure out not only the answer to your problem, but how to do these types of problems in general.

 

So, you need to find the 4th root of 1296, which means you need to find two or more numbers which, in addition to multiplying together to make 1296, are also perfect 4ths themselves.  In other words, we need to simplify the expression by first expanding the number into its component factors, choosing factors that work for our purposes.

 

So your first step might be to find the smallest perfect 4th you can and see if it divides evenly into 1296.  I chose 2⁴, which is 2 * 2 * 2 * 2, or 16.  Dividing 1296 by 16 gives you 81, which definitely looks promising.  (If 16 didn't divide evenly into 1296, I would have tried 3⁴ and continued up from there until I found one that worked.)

 

Next step: 81 breaks down into 9², and 9 is already a perfect square, so we're on the right track.  81 = 3 * 3 * 3 * 3, or 3⁴.  Perfect!

 

So our reformatted expression is:

 

4th root of (2⁴ * 3⁴)

 

And the 4ths cancel out the root (since we got each term to have a 4th power in it - don't do this until each term is a 4th power) so we're left with just:

 

2 * 3

 

So your answer would be 6.

 

Hope this helps!  Just remember, you need to factor the number into terms that are also perfect powers to match the original root.  That way, the problem ends up saying "Take that two and that three, raise each to the power of 4, and then take the 4th root of each."  Roots and exponents cancel each other out, but only if they're the same power.  (If they're different powers, you can still simplify a bit, but you won't be able to completely clear both the root and the exponent all at once.)

Answer 2

1296=36²; 36=6² Thus 1296=(6²)²=6⁴;

 

So, ⁴√1296=6