Question 1204752
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Answer: <font color=red size=4>Choice C)  6k^2</font>


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Work Shown



Method 1
{{{6^(4x+3) = 6^(2(2x+1)+1)}}}


{{{6^(4x+3) = 6^(2(2x+1))*6^1}}}


{{{6^(4x+3) = (6^(2x+1))^2*6}}}


{{{6^(4x+3) = k^2*6}}}


{{{6^(4x+3) = 6k^2}}}
I reached this independently of ikleyn, so there's no citation needed.


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Method 2
{{{6^(2x+1) = k}}}


{{{log(6,(k)) = 2x+1}}}


{{{2x+1 = log(6,(k))}}}


{{{2x = log(6,(k))-1}}}


{{{x = 0.5log(6,(k)) - 0.5}}}


{{{6^(4x+3) = 6^(4*(0.5log(6,(k)) - 0.5)+3)}}}


{{{6^(4x+3) = 6^(2*log(6,(k)) - 2+3)}}}


{{{6^(4x+3) = 6^(log(6,(k^2)) + 1)^""}}}


{{{6^(4x+3) = 6^(log(6,(k^2)))^""*6^1}}}


{{{6^(4x+3) = k^2*6}}}


{{{6^(4x+3) = 6k^2}}}


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Method 3


Let {{{k = f(x) = 6^(2x+1)}}}
Use graphing software like Desmos or GeoGebra to graph each of the five answer choices A through E as separate functions. 
Some teachers will allow graphing tech like that during exams, so it depends on the teacher. Often the app is set to exam mode to be for limited cases. Meaning you won't have access to the CAS feature for instance.


Anyways, set up f(x) = 6^(2x+1) in the first box.
12k for instance would be 12*f(x), k^2+6 would be (f(x))^2 + 6, and so on.
There would be 6 functions defined so far.
As the seventh function, plot 6^(4x+3) to see what curve perfectly aligns to it.
It should be 6*( f(x) )^2


Here's an example comparing answer choice B to 6^(4x+3)
<a href = "https://www.desmos.com/calculator/kivtpkjz6r">https://www.desmos.com/calculator/kivtpkjz6r</a>
The curves do not overlap, so we can rule choice B out.


However, if we plot choice C and 6^(4x+3) together, then we get this perfect overlap
<a href = "https://www.desmos.com/calculator/xbdq1rhx20">https://www.desmos.com/calculator/xbdq1rhx20</a>
Click the round button for graph 7 to turn it off, then click it back on. Repeatedly do this to have the curve blink two different colors to show the overlap.


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Method 4


We'll use graphing software similar to method 3.
Define f(x) = 6^(2x+1)
Then
A(x) = 12*f(x)
B(x) = (f(x))^2+6
C(x) = 6*(f(x))^2
D(x) = 2*f(x)+6
E(x) = 36*(f(x))^2
are the five answer choices, and we want to check against
G(x) = 6^(4x+3)
To see if two function curves are the same, we can subtract them and see which produces a horizontal flat line over the x axis.


Through trial-and-error, you should find that C(x) - G(x) will produce that flat line we're after.
It means C(x) - G(x) = 0 for all x in the domain, which leads to C(x) = G(x)


It's the same idea as saying something like 2+3 = 5, so 2+3-5 = 0.



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Method 5


Use the route that tutor Edwin took. 
It's a useful strategy for exams involving multiple choice (when in a time crunch and/or when completely stumped).
x = 0 is a common easy value to work with. It leads to 6^(4x+3) = 216 and k = 6. 
You can then plug that k value into each answer choice to see which gives you 216. 
If you had more than one answer choice produce 216, then pick something else for x.
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