Hey all. I'm trying to solve this question, but don't really understand how to do it:

If (x^4)^5 = x^a for all values of 'x', what is the value of 'a'?

Any help is much appreciated, thanks!

By the way, I've gotten to x^20 = x^a, but don't know what to do from here.

suppose I said

x*20 = x*a

would you not say a=20?

Same here. Two powers of x are the same if the exponents are the same.

Or, if you've ventured into logs, then you have

20 logx = a logx
divide by logx and you have
20 = a

This only fails if x=1, since then logx = 0, and you cannot divide by 0. Plus, since 1^a = 1 for any value of a, then 1^20 = 1^a does not mean that a=20.

Thank you for the explanation Steve! I understand why I was confused and get the problem now. Again, thanks!

Sure! To find the value of 'a', we need to understand the properties of exponents.

In this case, we have the expression (x^4)^5. According to the exponent rule of raising a power to a power, we can multiply the exponents together. So, (x^4)^5 is equal to x^(4*5), which simplifies to x^20.

Now, we want to determine when x^20 is equal to x^a for all values of 'x'. For this to happen, the exponents must be equal. Therefore, a must be equal to 20.

Hence, the value of 'a' is 20.