If \(5^x5^{x3}=124\cdot 5^y,\) what is y in terms of \(x?\)
A. \(x\)
B. \(x  6\)
C. \(x  3\)
D. \(2x + 3\)
E. \(2x + 6\)
[spoiler]OA=C[/spoiler]
Source: GMAT Prep
If \(5^x5^{x3}=124\cdot 5^y,\) what is y in terms of \(x?\)
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$$5^x5^{x3}=124\cdot5^y$$
$$factorizing\ 5^x5^{x3}$$
$$5^{x3}\cdot\left(5^31\right)=124\cdot5^y$$
$$5^{x3}\cdot\left(251\right)=124\cdot5^y$$
$$x3=y$$
$$y=x3$$
$$Answer\ =\ C$$
$$factorizing\ 5^x5^{x3}$$
$$5^{x3}\cdot\left(5^31\right)=124\cdot5^y$$
$$5^{x3}\cdot\left(251\right)=124\cdot5^y$$
$$x3=y$$
$$y=x3$$
$$Answer\ =\ C$$
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 Brent@GMATPrepNow
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One option is to rewrite the left side of the equation by factoring out 5^(x3)Gmat_mission wrote: ↑Wed Jun 24, 2020 8:12 amIf \(5^x5^{x3}=124\cdot 5^y,\) what is y in terms of \(x?\)
A. \(x\)
B. \(x  6\)
C. \(x  3\)
D. \(2x + 3\)
E. \(2x + 6\)
[spoiler]OA=C[/spoiler]
Source: GMAT Prep
So, we get: 5^(x3)[5^3  1] = (124)(5^y)
Evaluate to get: 5^(x3)[125  1] = (124)(5^y)
Simplify to get: 5^(x3)[124] = (124)(5^y)
Divide both sides by 124 to get: 5^(x3) = 5^y
So, x3 = y
Answer: C

ASIDE: A lot of students struggle to see how we can factor 5^x  5^(x3) to get 5^(x3)[5^3  1]
Sure, they may be okay with straightforward factoring like these examples:
k^5  k^3 = k^3(k^2  1)
m^19  m^15 = m^15(m^4  1)
But they have problems when the exponents are variables.
IMPORTANT: Notice that, each time, the greatest common factor of both terms is the term with the smaller exponent.
So, in the expression 5^x  5^(x3), the term with the smaller exponent is 5^(x3, so we can factor out 5^(x3)
Likewise, w^x + x^(x+5) = w^x(1 + w^5)
And 2^x  2^(x2) = 2^(x2)[2^2  1]

Cheers,
Brent
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 Brent@GMATPrepNow
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Another option is to PLUG IN a value for x and see what kind of relationship we get between x and y.Gmat_mission wrote: ↑Wed Jun 24, 2020 8:12 amIf \(5^x5^{x3}=124\cdot 5^y,\) what is y in terms of \(x?\)
A. \(x\)
B. \(x  6\)
C. \(x  3\)
D. \(2x + 3\)
E. \(2x + 6\)
[spoiler]OA=C[/spoiler]
Source: GMAT Prep
There are two "nice" xvalues to plug in. They are x = 0 and x = 3, since we can easily use these to evaluate 5^x and 5^(x3). Of these two values, x = 3 is the easier one to plug in.
So, let's plug in x = 3
We get: 5^3  5^(33) = (124)(5^y)
Simplify to get: 125  1 = (124)(5^y)
Simplify to get: 124 = (124)(5^y)
Divide both sides by 124 to get: 1 = 5^y
Solve for y to get: y = 0
So, when x = 3, y = 0.
Now we'll check the answer choices to see which one satisfies this relationship.
A) y = x... So, we get 0 = 3 (NOPE)
B) y = x  6... So, we get 0 = 3  6 (NOPE)
C) y = x  3... So, we get 0 = 3  3 IT WORKS!
D) y = 2x + 3... So, we get 0 = 2(3) + 3 (NOPE)
E) y = 2x + 6... So, we get 0 = 2(3) + 6 (NOPE)
Answer: C
Cheers,
Brent
Brent Hanneson  Creator of GMATPrepNow.com
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 Scott@TargetTestPrep
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Solution:Gmat_mission wrote: ↑Wed Jun 24, 2020 8:12 amIf \(5^x5^{x3}=124\cdot 5^y,\) what is y in terms of \(x?\)
A. \(x\)
B. \(x  6\)
C. \(x  3\)
D. \(2x + 3\)
E. \(2x + 6\)
[spoiler]OA=C[/spoiler]
We can simplify the given equation:
5^x – 5^(x 3) = 124*5^y
5^x – (5^x)(5^ 3) = 124*5^y
5^x(1 – (5^ 3)) = 124*5^y
5^x(1 – 1/125) = 124*5^y
5^x(1 – 1/125) = 124*5^y
5^x(124/125) = 124*5^y
5^x = 124*5^y*(125/124)
5^x = 5^y(125)
5^x = 5^y(5^3)
5^x = 5^y+3
x = y + 3
x  3 = y
Answer: C
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