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GMAT prep Exponent question

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Junior | Next Rank: 30 Posts
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by batman73 » Sun Mar 15, 2009 8:38 pm
Hi,
Can anyone help me solve this problem. For some reason I am not seeing the method for solving this problem. The answer=20, but the steps to take to solve the problem is what I need help with

If (3^x)- (3^x-1)=162, then x(x-1)=

c) 20 <--Answer
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Re: GMAT prep Exponent question

by Stuart@KaplanGMAT » Sun Mar 15, 2009 9:19 pm
batman73 wrote:Hi,
Can anyone help me solve this problem. For some reason I am not seeing the method for solving this problem. The answer=20, but the steps to take to solve the problem is what I need help with

If (3^x)- (3^x-1)=162, then x(x-1)=

c) 20 <--Answer
I'd attack it using some logic and common sense.

We know that 3^x and 3^(x-1) are consecutive powers of 3. We also know that they're 162 apart.

I'd just list powers of 3:

3
9
27
81
243

Ding! 243-81 = 162.

243 is 3^5 and 81 is 3^4, so x(x-1) = 5*4 = 20
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Re: GMAT prep Exponent question

by Vemuri » Mon Mar 16, 2009 12:42 am
batman73 wrote:Hi,
Can anyone help me solve this problem. For some reason I am not seeing the method for solving this problem. The answer=20, but the steps to take to solve the problem is what I need help with

If (3^x)- (3^x-1)=162, then x(x-1)=

c) 20 <--Answer
You can solve this very easily using the exponent rules.

3^x - 3^(x-1) = 162
==> 3^(x-1) * [3-1] = 162
==> 3^(x-1) * 2 = 162
==> 3^(x-1) = 81
==> 3^(x-1) = 3^4 (3^4 is equal to 81)
==> x-1 = 4 (since the base of the exponents is the same)
==> x = 5

Hence, x(x-1) = 5*4 = 20

Hope this helps.
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by KingA » Mon Mar 16, 2009 10:30 am
3^x - 3^(x-1) = 162
==> 3^(x-1) * [3-1] = 162
Can you explain this step further? What is the rule here?
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by awesomeusername » Mon Mar 16, 2009 10:46 am
For Example:

3^5 - 3^4 = 3^4(3^1 - 1) = 3^4(2)

So:

3^X - 3^(X-1) = 3^(X-1)(3^1 - 1) = 3^(X-1)*2

You're factoring out a SMALLER power with the SAME base.
Constant dripping hollows out a stone.
-Lucretius
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explanation of the exponent rule

by kapitalist » Tue Mar 17, 2009 2:44 pm
I had a similar question to this and the people here were a great help but some people who read still dont get (i forwarded some people to this page) because the rule is not explained.

in 3^k - 3^k-1, the reason it factors out is because when you multiply numbers with the same base you add the exponents...

so the reason 3^k-1(3-1) is so is because its like saying

3^k-1(3^1 - 1) <<<< Do you see that you would add the exponents

k-1+1 because both bases are 3. Thats the part I think people get hung up on and I hope this little piece of extra explanation helps.
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