SOLUTION: what is the value of x? 4^20+4^20=2^x

Algebra ->  Exponents -> SOLUTION: what is the value of x? 4^20+4^20=2^x      Log On


   

Question 177704: what is the value of x?
4^20+4^20=2^x
Answer by EMStelley(208) About Me  (Show Source):
You can put this solution on YOUR website!
To solve this problem you need to be a little clever. The first thing to notice is that everything on the left hand side is a power of 4 while everything on the right hand side is a power of 2. It would be nice if they were the same, specifically 2, since that's the side with the variable. Can we change 4%5E20 into base 2? Sure, since 4=2%5E2. So we have that
%282%5E2%29%5E20=4%5E20
And because of the rules of exponents, we have that
2%5E40=4%5E20
So we can rewrite the equation:
2%5E40%2B2%5E40=2%5Ex
Now, we combine like terms on the left hand side:
2%282%5E40%29=2%5Ex
What is 2%282%5E40%29? Well, all we're doing is multiplying by another 2, so it's just 2%5E41. So,
2%5E41=2%5Ex
So x=41.