alltimeacheiver wrote:

If \(5^x – 5^{(x- 3)} = 124*5^y\), what is y in terms of x?

A. x

B. x - 6

C. x - 3

D. 2x + 3

E. 2x + 6

One option is to rewrite the left side of the equation by factoring out 5^(x-3)

So, we get: 5^(x-3)[5^3 - 1] = (124)(5^y)

Evaluate to get: 5^(x-3)[125 - 1] = (124)(5^y)

Simplify to get: 5^(x-3)[124] = (124)(5^y)

Divide both sides by 124 to get: 5^(x-3) = 5^y

So, x-3 = y

Answer: C

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ASIDE: A lot of students struggle to see how we can factor 5^x - 5^(x-3) to get 5^(x-3)[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^(x-3), the term with the

smaller exponent is 5^(x-3, so we can factor out 5^(x-3)

Likewise, w^x + x^(x+5) = w^x(1 + w^5)

And 2^x - 2^(x-2) = 2^(x-2)[2^2 - 1]

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Cheers,

Brent

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