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If 5^x 5^(x- 3) = 124*5^y, what is y in terms of x?
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Updated on: 18 Sep 2022, 07:59
Updated on: 18 Sep 2022, 07:59
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Question Stats:
77% (01:42) correct
22% (00:58) wrong based on 9 sessions
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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
A. x
B. x - 6
C. x - 3
D. 2x + 3
E. 2x + 6
Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
Re: If 5^x 5^(x- 3) = 124*5^y, what is y in terms of x?
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21 Jul 2020, 09:53
21 Jul 2020, 09:53
1
Carcass 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
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]
------------------------------------------------------------
Cheers,
Brent
Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
Re: If 5^x 5^(x- 3) = 124*5^y, what is y in terms of x?
[#permalink]
21 Jul 2020, 09:53
21 Jul 2020, 09:53
1
Carcass 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
A. x
B. x - 6
C. x - 3
D. 2x + 3
E. 2x + 6
Another option is to PLUG IN a value for x and see what kind of relationship we get between x and y.
There are two "nice" x-values to plug in. They are x = 0 and x = 3, since we can easily use these to evaluate 5^x and 5^(x-3). 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^(3-3) = (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
