Problem Solving

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. I appreciate any help I can get in explaining this one through.

If 4^a + (4^a+1) = (4^a+2) - 176, what is the value of a?

If 4^a + (4^a+1) = (4^a+2) - 176, what is the value of a?

Two ways to think about this. The first, if I see that the algebra will be a little hairy, is to work with values. Let's see what happens when I raise 4 to various exponents:

4^1 = 4
4^2 = 16
4^3 = 64
4^4 = 256
4^5 = 1024

Well, the question tells me that if I add two if these numbers together, it will be equal to the next number in the list minus 176. Clearly, it doesn't work if a = 1 as 4^1 + 4^2 is not 4^3 - 176. (4 + 16 is obviously not 64 - 176.)

But if a = 2, we get 4^2 + 4^3 = 4^4 - 176. (16 + 64 = 256 - 176; or 80 =80.) Success! a = 2.

The alternative, of course, is to do this algebraically. This is somewhat less fun.

If 4^a + (4^a+1) = (4^a+2) - 176, what is the value of a?

First, let's get all the a's on one side.

4^a + (4^a+1) - (4^a+2) = -176

One could rewrite as 4^a + (4^a * 4^1) - (4^a *4^2) = -176

Factor out the 4^a to get

4^a * (1 + 4^1 - 4^2) = - 176

4^a * (1 + 4 - 16) = - 176

4^a * (- 11) = - 176

4^a = 16

4^a = 4^2

a=2.

gouldimal wrote: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. I appreciate any help I can get in explaining this one through.

If 4^a + (4^a+1) = (4^a+2) - 176, what is the value of a?

4^(a+2) - 4^(a+1) - 4^a = 176.
In other words:
176 = difference between 3 CONSECUTIVE POWERS OF 4.

List the powers of 4:
4, 16, 64, 256...
The next greatest power of 4 will be too great (over 1000).
Test the 3 greatest values in the list above:
256-64-16 = 176.
Success!

Since 4^a = 16, a=2.

We could also do this, which seems easiest.

4ᵃ�² - 4ᵃ�¹ - 4ᵃ = 176

or

4ᵃ * (4² - 4 - 1) = 176

or

4ᵃ * 11 = 176

or

4ᵃ = 16

so a = 2, and we're set.