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505-555 Level|   Algebra|   Exponents|      
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Bunuel
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If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2? 23 Sep 2015, 22:35
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C
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78% (01:38) correct 22% (01:48) wrong based on 369 sessions

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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


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C
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BrentGMATPrepNow
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If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2? Updated on: 22 Jul 2022, 07:42
Originally posted by BrentGMATPrepNow on 24 Sep 2015, 07:18.
Last edited by BrentGMATPrepNow on 22 Jul 2022, 07:42, edited 1 time in total.
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Bunuel
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.

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

Factor out the greatest common divisor: \(5^{x-1}(5^1 - 1)= 500\)

Evaluate the part in brackets: \(5^{x-1}(4)= 500\)

Divide both sides by \(4\) to get: \(5^{x-1}= 125\)

Rewrite the right side as a power of \(5\) to get: \(5^{x-1}= 5^3\)

Since the bases are equal, we can conclude that: \(x-1= 3\), which means \(x = 4\)

If \(x = 4\), then \((x - 1)^2 = (4 - 1)^2 = 3^2 = 9\)

Answer: C
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If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2? 23 Sep 2015, 23:55
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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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If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2? 24 Sep 2015, 01:45
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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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If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2? 24 Sep 2015, 02:28
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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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If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2? 24 Sep 2015, 06:06
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Bunuel
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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If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2? 24 Sep 2015, 07:10
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DavidSt
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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If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2? 24 Sep 2015, 10:09
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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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If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2? 24 Sep 2015, 11:40
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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

Final Answer:
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C

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If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2? 24 Sep 2015, 13:22
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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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If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2? 24 Sep 2015, 18:28
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DavidSt
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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If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2? 25 Sep 2015, 23:12
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Bunuel
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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If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2? 27 Sep 2015, 00:27
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(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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If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2? 22 Jul 2022, 18:55
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Bunuel
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

Notice that 5^x = 5 * 5^(x - 1). Thus, we can write:

\(\Rightarrow\) 5^x - 5^(x - 1) = 500

\(\Rightarrow\) 5 * 5^(x - 1) - 5^(x - 1) = 500

\(\Rightarrow\) 5^(x - 1)[5 - 1] = 500

\(\Rightarrow\) 5^(x - 1) * 4 = 500

\(\Rightarrow\) 5^(x - 1) = 125

\(\Rightarrow\) 5^(x - 1) = 5^3

Equating the exponents of the expressions on each side of the equation, we get x - 1 = 3, which means x = 4. Thus, (x - 1)^2 = 3^2 = 9.

Answer: C
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If 5^x - 5^{x-1}= 500 , what is the value of (x - 1)^2? 15 Jan 2023, 03:37
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Given that \(5^x - 5^{x-1}= 500\) and we need to find the value of (x - 1)^2

\(5^x - 5^{x-1}\) = 500
=> \(5^{x-1 + 1} - 5^{x-1}\) = 500
=> \(5^{x-1} * 5^1 - 5^{x-1}\) = 500
=> \(5^{x-1} * (5 - 1)\) = 500
=> \(5^{x-1} = \frac{500}{4}\) = 125 = \(5^3\)
=> x - 1 = 3
=> x = 4

=> \((x - 1)^2\) = \((4 - 1)^2\) = 9

So, Answer will be C
Hope it helps!

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