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Exponent Rules

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by somail » Thu Aug 28, 2008 12:13 pm
How is the following simplification done:

2^100-2^96
2^96*(2^4-1) (This is the step I do not get)
2^96*(16-1)
15*2^96
2^96*3*5.


How is 2^11 turned into 2^(4-1) and then multiplied against 2^96? Is there a general rule?

thanks
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Source: — Problem Solving |

Re: Exponent Rules

by Stuart@KaplanGMAT » Thu Aug 28, 2008 12:27 pm
somail wrote:How is the following simplification done:

2^100-2^96
We can only add or subtract in these situations if the base is the same AND the exponent is the same.

For example, we can solve:

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

However, there's no simple solution for:

5(x^5) - 2(x^3)

So, if we're faced with a situation in which the bases are the same but the exponents aren't, we need to find a common exponent. We do so through factoring.

Let's look at the original question:

2^100 - 2^96

To get the same exponent, we need to reduce 2^100.

We can rewrite 2^100 as (2^4)(2^96) = 16(2^96).
We can rewrite 2^96 as 1(2^96).

So, we can rewrite 2^100 - 2^96 as:

16(2^96) - 1(2^96) = 15(2^96)
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by somail » Thu Aug 28, 2008 12:43 pm
Wow. Thank you so much. Thats been bugging me all morning.
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