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# If 625^(-x) + 25^(-2x) + 5^(-4x) = 15, what is the value of x?

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Math Expert
Joined: 02 Sep 2009
Posts: 50670
If 625^(-x) + 25^(-2x) + 5^(-4x) = 15, what is the value of x?  [#permalink]

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21 May 2015, 04:26
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4
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Difficulty:

15% (low)

Question Stats:

88% (01:45) correct 12% (02:18) wrong based on 149 sessions

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If $$625^{(-x)} + 25^{(-2x)} + 5^{(-4x)} = 15$$, what is the value of x?

A. -4
B. -1/4
C. 0
D. 1/4
E. 4

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Posts: 63
Re: If 625^(-x) + 25^(-2x) + 5^(-4x) = 15, what is the value of x?  [#permalink]

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21 May 2015, 04:58
1
625^(-x) + 25^(-2x) + 5^(-4x) = 15

i.e.

(1/625)^x + (1/25)^(2x) + (1/5)^(4x) = 15

(1/625)^x + (1/25^2)^x + (1/5^4)^x = 15

(1/625)^x +(1/625)^x + (1/625)^x = 15

3 * (1/625)^x =15

(1/625)^x = 5

(1/5)^4x = 5

5^-4x = 5^1

Equating powers :

-4x =1 ; x =-1/4

Ambarish
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Re: If 625^(-x) + 25^(-2x) + 5^(-4x) = 15, what is the value of x?  [#permalink]

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21 May 2015, 05:05
1
Bunuel wrote:
If 625^(-x) + 25^(-2x) + 5^(-4x) = 15, what is the value of x?

A. -4
B. -1/4
C. 0
D. 1/4
E. 4

Simplify the stem to 1/625^(-x)+1/625^(-x)+1/625^(-x)=15

5^4 = 625 so 625^(1/4) = 5 but currently 625 is on the denominator so we need (-1/4).

i.e 625^(-1/4)+625^(-1/4)+625^(-1/4) = 5+5+5 =15

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Re: If 625^(-x) + 25^(-2x) + 5^(-4x) = 15, what is the value of x?  [#permalink]

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21 May 2015, 16:24
1
Simplifying 625^-x = 5^-4x, similarly 25^-2x=5^-4x; hence
we get : 5^-4x+5^-4x+5^-4x = 3.5 therefore 3*5^-4x=3.5
Comparing power f 5, we get -4x=1 therefore x=-1/4

Thanks
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Re: If 625^(-x) + 25^(-2x) + 5^(-4x) = 15, what is the value of x?  [#permalink]

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21 May 2015, 18:32
1
Bunuel wrote:
If 625^(-x) + 25^(-2x) + 5^(-4x) = 15, what is the value of x?

A. -4
B. -1/4
C. 0
D. 1/4
E. 4

My answer is B as well. Got there by simplifying to the same bases, as above!

625^(-x) + 25^(-2x) + 5^(-4x) = 15
becomes

(5^4)^-x) + (5^2)^(-2x) + 5^(-4x) = 5 * 3

becomes

5^-4x * 3 = 5 * 3

Now we can solve for x:
-4x = 1

x = -1/4
Math Expert
Joined: 02 Sep 2009
Posts: 50670
Re: If 625^(-x) + 25^(-2x) + 5^(-4x) = 15, what is the value of x?  [#permalink]

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25 May 2015, 06:17
1
Bunuel wrote:
If 625^(-x) + 25^(-2x) + 5^(-4x) = 15, what is the value of x?

A. -4
B. -1/4
C. 0
D. 1/4
E. 4

OFFICIAL SOLUTION:

625^(-x) + 25^(-2x) + 5^(-4x) = 15;

5^(-4x) + 5^(-4x) + 5^(-4x) = 15;

3*5^(-4x) = 15;

5^(-4x) = 5.

Bases are equal, so we can equate the powers: -4x = 1, giving x = -1/4.

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Re: If 625^(-x) + 25^(-2x) + 5^(-4x) = 15, what is the value of x?  [#permalink]

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26 May 2015, 21:10
Hi All,

This is a great 'concept' question; the answer choices are written in such a way that you can use them against the prompt and actually avoid most of the "math" altogether (although you do need to understand your Exponent rules....).

We're told that 625^(-X) + 25^(-2X) + 5^(-4X) = 15. We're asked for the value of X.

Since each of the calculated terms MUST be positive (regardless of what the exponent is), we can use the "bases" to our advantage.....

With Answer A, we'd have 625^4, which is MUCH BIGGER than 15 (and we'd be adding to that big number). Eliminate A.
With Answer E, we'd have 625^(-4), which would create a TINY fraction (and we'd add some other fractions to it, so the total would be MUCH TOO SMALL). Eliminate E.

With Answer D, we'd have 625^(-1/4), which will also be a fraction (just not as tiny as the one in answer E), but the total would still be TOO SMALL. Eliminate D.

With Answer C, anything to the '0 power' is 1, so we'd have 1+1+1 = 3. This is not 15. Eliminate C.

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Re: If 625^(-x) + 25^(-2x) + 5^(-4x) = 15, what is the value of x?  [#permalink]

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Re: If 625^(-x) + 25^(-2x) + 5^(-4x) = 15, what is the value of x? &nbs [#permalink] 25 Oct 2017, 04:54
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