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exponent problems

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by Stuart@KaplanGMAT » Tue Sep 02, 2008 12:33 pm
1 question per post per customer! Also, in future, please post in the problem solving or data sufficiency sub forum, as appropriate. Hey, please include the source and the answer choices as well!

(If you do a search on your second question, you'll see that it's been answered many times.)
if 2^x - 2^(x-2) = 3(2^13), what is x?


Let's start by focusing on the left side of the equation:
2^x - 2^(x-2)
There's no simple way to add or subtract exponents unless they have the same power AND the same base. So, we need to rewrite this expression so the power and base are equal. We do so by factoring down to the smaller power.

2^x - 2^(x-2)

We can rewrite 2^x as:

(2^2)(2^(x-2))
4(2^(x-2))

We can rewrite 2^(x-2) as:
1(2^(x-2))

So, we have:

4(2^(x-2)) - 1(2^(x-2)) = 3(2^(x-2))

Plugging that back into the original equation, we get:

3(2^(x-2)) = 3(2^13)

so, we know that:

x - 2 = 13

x = 15
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by mberkowitz » Tue Sep 02, 2008 12:45 pm
2^x - 2^(x-2)

We can rewrite 2^x as:

(2^2)(2^(x-2))
4(2^(x-2))

We can rewrite 2^(x-2) as:
1(2^(x-2))

can you please elaborate on the factoring down of the exponents? specifically I don't understand how you get 2^2... thanks and my apologies for not following the rules.
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by Stuart@KaplanGMAT » Tue Sep 02, 2008 1:02 pm
mberkowitz wrote:2^x - 2^(x-2)

We can rewrite 2^x as:

(2^2)(2^(x-2))
4(2^(x-2))

We can rewrite 2^(x-2) as:
1(2^(x-2))

can you please elaborate on the factoring down of the exponents? specifically I don't understand how you get 2^2... thanks and my apologies for not following the rules.
Let's look at a simpler example:

3(2^5) - (2^2) = ?

There's no easy way to subtract the two terms. We need to equalize the exponents. To do so, we always (well, almost always) reduce the larger power to the smaller one.

We have the general exponent rule:

x^a * x^b = x^(a+b).

So, we can rewrite 2^5 as 2^(3+2) = (2^3)*(2^2)

Subbing in for 2^5, we get:

3(2^3)(2^2) - (2^2)
3(8)(2^2) - 1(2^2)
24(2^2) - 1(2^2)
23(2^2)

In the actual question, we want to reduce to the exponent (x-2). I used the x^(a+b) rule to rewrite 2^x.

2^x = 2^(2 + x - 2) = 2^2 * 2^(x-2)
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