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If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2?

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If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2?  [#permalink]

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New post 23 Sep 2015, 21:35
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A
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C
D
E

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Question Stats:

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Re: If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2?  [#permalink]

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New post 23 Sep 2015, 22:55
1
5^x - (5^x/5) = 500

=> 5^x(1-1/5) = 500

=> 5^x * (4/5) = 500

=> 5^(x-1) = 125

=> x=4

hence (4-1)^2 = 9 hence C.

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Re: If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2?  [#permalink]

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New post 24 Sep 2015, 00:45
1
5^x - 5^(x-1) = 500
=> 5^(x-1) {5 - 1} =500
=> 5^(x-1) 4 = 500
=> 5^(x-1) 2^2 = 5^3 2^2
=> 5^(x-1) = 5^3
=> x-1 = 3
=> (x-1)^2 = 3^2 = 9

Answer: C
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Re: If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2?  [#permalink]

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New post 24 Sep 2015, 01:28
1
Can you not just work this out?

5^4 - 5^(4-1) = 625 - 125 = 500

Therefore x is 4 so (x-1)^2 = (4-1)^2 = 9 (c)

This way just seems quicker to me!
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Re: If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2?  [#permalink]

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New post 24 Sep 2015, 05:06
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Bunuel wrote:
If \(5^x - 5^{x-1}= 500\), what is the value of (x - 1)^2?

(A) 1
(B) 4
(C) 9
(D) 25
(E) 36


Kudos for a correct solution.


\(5^x - 5^{x-1}= 500\)

or \(5^x - \frac{5^x}{5}= 500\)

or \(5^x(1 - \frac{1}{5})= 500\)

or \(5^x( \frac{4}{5})= 125*4\)

or \(5^{x-1}\) = 125

or \(5^{x-1}= 5^3\)

or \(x-1 = 3\)

or \((x-1)^2 = 9\)

Answer:- C
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Re: If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2?  [#permalink]

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New post 24 Sep 2015, 06:10
DavidSt wrote:
Can you not just work this out?

5^4 - 5^(4-1) = 625 - 125 = 500

Therefore x is 4 so (x-1)^2 = (4-1)^2 = 9 (c)

This way just seems quicker to me!


How did you "work it out"?
If you guess and tested, that's totally fine, except that approach could take a while.

Cheers,
Brent
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Re: If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2?  [#permalink]

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New post 24 Sep 2015, 06:18
2
Bunuel wrote:
If \(5^x - 5^{x-1}= 500\), what is the value of (x - 1)^2?

(A) 1
(B) 4
(C) 9
(D) 25
(E) 36


Kudos for a correct solution.


Testing the answer choices could also be pretty quick.

A) 1
This means (x - 1)^2 = 1
So, x - 1 = 1, which mean x = 2
Now plug x = 2 into \(5^x - 5^{x-1}= 500\) to get \(5^2 - 5^{2-1}= 500\)
Simplify: 25 - 5 = 500
Doesn't work


B) 4
This means (x - 1)^2 = 4
So, x - 1 = 2, which mean x = 3
Now plug x = 3 into \(5^x - 5^{x-1}= 500\) to get \(5^3 - 5^{3-1}= 500\)
Simplify: 125 - 25 = 500
Doesn't work

C) 9
This means (x - 1)^2 = 9
So, x - 1 = 3, which mean x = 4
Now plug x = 4 into \(5^x - 5^{x-1}= 500\) to get \(5^4 - 5^{4-1}= 500\)
Simplify: 625 - 125 = 500
WORKS!

Answer: C

Cheers,
Brent
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Re: If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2?  [#permalink]

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New post 24 Sep 2015, 09:09
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I believe the answer is C. Please see below for explanation

If 5^x−5^x−1=500, what is the value of (x - 1)^2?

5^x(1 - 1/5 ) = 5* 2*5 * 2*5

5^x(4/5) = 5^3 * 2^2

5^x * 4 = 5^4 * 2^2

5^x * 2^2 = 5^4 * 2^2

Which means that x = 4

Replacing x with 4 in (x - 1)^2 gives 9


Answer C
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Re: If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2?  [#permalink]

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New post 24 Sep 2015, 10:40
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Hi All,

This question has a great 'brute force' shortcut to it (if you're comfortable doing some basic multiplication).

Since the answer choices are all perfect squares, X MUST be an integer.

We're also given 5^X and 5^(X-1), which are two consecutive "powers" of 5. We're told that subtracting the smaller value from the larger value will give us 500...

Let's start listing powers of 5 until we find two consecutive powers that differ by 500....

5^0 = 1
5^1 = 5
5^2 = 25
5^3 = 125
5^4 = 625

STOP. 625 - 125 = 500, so X MUST be 4. To confirm...

5^4 - 5^(4-1) =
5^4 - 5^3 =
625 - 125 = 500

Since we now know the value of X, we can answer the question - the value of (X-1)^2 = (4-1)^2 = 9

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If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2?  [#permalink]

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New post 24 Sep 2015, 12:22
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\(5^x−5^{x-1}\)=500

This can be simplified to

\(5^x\) * (4/5) =\(5^3\) * \(2^2\)


\(5^{x-1}\) * \(2^2\) = \(5^3\) * \(2^2\)

x - 1 = 3


\((x-1) ^ 2\) = 9

Answer = C
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Re: If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2?  [#permalink]

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New post 24 Sep 2015, 17:28
GMATPrepNow wrote:
DavidSt wrote:
Can you not just work this out?

5^4 - 5^(4-1) = 625 - 125 = 500

Therefore x is 4 so (x-1)^2 = (4-1)^2 = 9 (c)

This way just seems quicker to me!


How did you "work it out"?
If you guess and tested, that's totally fine, except that approach could take a while.

Cheers,
Brent


To be honest I have tried to memorize the lowest numbers and their exponents up to around 6-8 (depending on the number i.e. 2^8 but not 5^8) as that was a tip from the Manhattan Guides.

As soon as I saw 5^x I just thought: 5, 25, 125, 625 and instantly noticed that 625-125 = 500.

But I can see the merit in the other way of factoring as the questions may not all have an easy solution like this one!
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Re: If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2?  [#permalink]

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New post 25 Sep 2015, 22:12
1
Bunuel wrote:
If \(5^x - 5^{x-1}= 500\), what is the value of (x - 1)^2?

(A) 1
(B) 4
(C) 9
(D) 25
(E) 36


Kudos for a correct solution.


Simplify both sides to their basic roots

\(5^x - 5^{x-1}= 500\)

L.H.S R.H.S
\(5^{x-1}(5-1)\) = \(5^3 * 2^2\)
\(5^{x-1}(4)\) = \(5^3 * 2^2\)
\(5^{x-1}(4)\) = \(5^3 * 2^2\)

Therefore
x-1 = 3
Square both sides
\((x-1)^2\) = 9

Cheers
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Re: If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2?  [#permalink]

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New post 26 Sep 2015, 23:27
(5^x) - (5^x)*5 = 500
5^x (1-5) = 500
5^x = -125
5^x = -5^3
x=3 => (3-1)^2 = 4
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Re: If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2?  [#permalink]

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Re: If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2?   [#permalink] 06 Jan 2020, 04:21
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