Algebra Strategy Guide, Chp. 3 - Exponents

[clarence.booth]

Hi,

I'm need help on working though the logic of a problem in the Alg. Strat, Guide, #8 on pg. 45:

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

I'm not sure exactly what the initial point of entry is for solving the problem. Please advise.

Thanks,

Clarence

[anshul04]

Clarence,

For most of the problems involving exponents, I normally take one basic approach - simplify the given equation / expression to consolidate common base(s) (by adding / subtracting / multiplying exponents) so that I have the given expression / equation in its most simplified form. Once I have that, it's just a matter of few seconds to figure out the right answer.

This problem also can be easily solved with this approach, as shown below. My entry point for this problem would be, as I mentioned above, to simplify the equation.

4^a + 4^a+1 = 4^a+2 - 176
4^a+2 - 4^a - 4^a+1 = 176
4^a.4^2 - 4^a - 4^a.4^1 = 4 x 4 x 11
4^a(4^2 - 1 - 4) = 4^2 x 11
4^a(16 - 1 - 4) = 4^2 x 11
4^a x 11 = 4^2 x 11

This gives me a = 2.

[tim]

Thanks, Anshul! Just to clarify, the appropriate strategy here is to factor out a 4^a from all of the powers of 4, leaving only numbers inside the parentheses.

[clarence.booth]

Thanks, Tim and Anshul.

One more question: How did we get the 1 in the (16-1-4)? Or in other words, how does the -4^a - 4^a become 4^0 = 1? When I look at it I want to convert it to -8^a.

What rule am I missing?

[tim]

I don't see where -4^a - 4^a has been turned into 4^0; can you point out where that has been done? Also, 16-1-4 was never turned into 1; it is 11, as indicated in the work above.

[TonyS610]

I thought the same thing at first. You have to factoring out 4^a, not turning 4^a-4^A in 0.

Hope that helps

[RonPurewal]

the point is that all of these powers of 4 can be turned into like terms.

4^a is 4 x 4 x 4 x ... x 4 (repeating 'a' times)

4^(a + 1) is 4 x 4 x 4 x ... x 4 (repeating 'a + 1' times)
which is just 'a' times, multiplied by another 4.
so, 4^(a + 1) = 4 • 4^a.

4^(a + 2) is 4 x 4 x 4 x ... x 4 (repeating 'a +21' times)
which is just 'a' times, multiplied by two more 4's.
so, 4^(a + 2) = 16 • 4^a.

once you've done this, you can combine all of these terms, since all of them are different multiples of 4^a.