Question 56·200 Super-Hard SAT Math Questions·Advanced Math

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The function $g$ is defined by

g(x)=10(12)^{x/2}(18)^{x/3}.

For every increase of $\frac{2}{3}$ in the value of $x$ , the value of $g(x)$ increases by a factor of $c$ , where $c$ is a constant.

Which choice rewrites $g(x)$ so that the value of $c$ appears as the base of the exponential expression?

An exponential factor belongs to a particular input interval. If a question asks for the factor over a fractional step, divide $x$ by that step to count how many times the factor applies. Desmos can find the factor by comparing outputs one step apart; exponent rules give its exact form. An equivalent function can still show the factor for the wrong interval as its base.

Hints

Hint 1
An exponent can count intervals rather than single units. If one interval is $\frac{2}{3}$ , how many such intervals fit into an increase of $x$ ?
Hint 2
A growth factor is a ratio of outputs, not their difference. Compare $g(\frac{2}{3})$ with $g(0)$ to find what multiplies the output over one requested step.
Hint 3
For an exact base, rewrite $12$ and $18$ as products of powers of $2$ and $3$ . Then multiply exponents when raising a power to a power, and add exponents only when the bases match.

Step-by-step

Find the factor for the requested interval
Step 1
Count the intervals in the exponent
At $x=0$ , both powers in the given function equal $1$ , so the starting output is $10$ :
$g(0)=10.$
A factor multiplies the output once per specified interval. Here an interval is $\frac{2}{3}$ , so $x$ contains $x/(2/3)$ intervals. Write that relationship:
$g(x)=10c^{x/(2/3)}.$
Divide by the fraction to make the interval count easier to read:
$g(x)=10c^{3x/2}.$
Step 2
Measure one interval's factor in Desmos
Type the given function as $g(x)=10(12)^{x/2}(18)^{x/3}$ , then type $g(2/3)/g(0)$ . Desmos shows about $4.35$ . The inputs differ by exactly one $\frac{2}{3}$ -unit interval, so their output ratio is $c$ . Keep the expression exact for the answer choices.
Step 3
Write that ratio exactly
Substitute $x=\frac{2}{3}$ into the given function and divide by $g(0)=10$ :
$c=\frac{10(12)^{(2/3)/2}(18)^{(2/3)/3}}{10}.$
Cancel the starting value $10$ :
$c=(12)^{(2/3)/2}(18)^{(2/3)/3}.$
Simplify each exponent:
$c=12^{1/3}18^{2/9}.$
Step 4
Rewrite the factor in the choices' exact form
Since $12=2^2\cdot3$ and $18=2\cdot3^2$ , replace their bases:
$c=(2^2\cdot3)^{1/3}(2\cdot3^2)^{2/9}.$
Give each factor its outside exponent; for a power of a power, multiply the exponents:
$c=2^{2/3}3^{1/3}2^{2/9}3^{4/9}.$
Add exponents only for matching bases:
$c=2^{2/3+2/9}3^{1/3+4/9}.$
Express both exponents in ninths:
$c=2^{8/9}3^{7/9}.$
Pull out their shared exponent:
$c=(2^8\cdot3^7)^{1/9}.$
Step 5
Put the requested factor in the base
Substitute this exact $c$ into the interval-count form from step 1:
$g(x)=10\left((2^8\cdot3^7)^{1/9}\right)^{3x/2}.$
The base is the factor for one $\frac{2}{3}$ -unit increase because the exponent rises by $1$ over that interval. This is the requested rewrite. Choice C.

Question 56·200 Super-Hard SAT Math Questions·Advanced Math

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