GMAT prep Exponent question

[batman73]

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

[Stuart@KaplanGMAT]

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

[Vemuri]

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.

[KingA]

3^x - 3^(x-1) = 162
==> 3^(x-1) * [3-1] = 162

Can you explain this step further? What is the rule here?

[awesomeusername]

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.

[kapitalist]

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.