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
