Value of M

[gouldimal]

Please help! I am working through the MGMAT Algebra book, and I am struggling with an exponents problem. MGMAT includes answers and step-by-step work, but I'm having a hard time following the logic. Thanks in advance.


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

[DavidG@VeritasPrep]

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.

[Brent@GMATPrepNow]

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

First notice that 2^18 = (2²)^9
Also notice that 20^n = (4 x 5)^n = (4^n)(5^n)

So, we can take (2^18)(5^m) = (20^n) and rewrite it as:
[(2²)^9][5^m] = (4^n)(5^n)
Simplify 2² to get: [4^9][5^m] = (4^n)(5^n)

From this, we can see that m = n [since 5^m must equal 5^n]
And we can see that n = 9, [since 4^9 must equal 4^n]

So, we get n = m = 9

So, [spoiler]m = 9[/spoiler]

Cheers,
Brent

[Matt@VeritasPrep]

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.