If m and n are positive integers and (2^18) (5^m) = (20^n), what is the value of m?

The key with these types of problems is often to take the prime factorization of every non-prime base. In this case that means breaking down 20. 20 = (2^2) * 5. This allows us to rewrite the equation as follows:

(2^18) (5^m) = [(2^2) * 5]^n

Next, we can distribute the 'n' on the right side of the equation. This gives us the following:

(2^18) (5^m) = (2^2n) * 5^n

If the exponents are integers, then equivalent bases must be raised to the same exponent on either side of the equation. In other words, we must have the same number of 2's and 5's on each side.

So we know that if we have 2^18 on one side and 2^2n on the other, it must be true that 18 = 2n, or n = 9

And if we have 5^m on one side and 5^n on the other, it must be true that m = n.

Well, if n=9 and m=n, m = 9.

Here's an approach:

2¹� * 5� = 20�

We want to break these down to prime bases, so let's start there.

2¹� * 5� = (2*2*5)�

Then we'll apply our exponent rules.

2¹� * 5� = 2�2�5�

Since we must have the same power on each prime base, we know n + n = 18, so n = 9. Since n = m (they're each the power on 5), we know m = 9.