Question 14·200 Super-Hard SAT Math Questions·Advanced Math

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For the exponential function $f(x)=\left(\frac{1}{8}\right)^{4-\frac{5}{3}x}$ , which choice rewrites $f(x)$ in the form $f(x)=c(32)^x$ , where $c$ is the minimum value of $f(x)$ for $x\ge 0$ ?

For an exponential function with a variable exponent, a base below $1$ doesn't automatically mean decay: check whether the exponent rises or falls as $x$ rises. Find the minimum on the allowed inputs, then evaluate the function there in Desmos. Rewrite the original base as a power of $2$ to put the function in the requested form.

Hints

Hint 1
The base is the number being raised to a power. For a base between $0$ and $1$ , lowering the exponent makes the result larger. As $x$ increases here, does the exponent rise or fall?
Hint 2
In the form $c(32)^x$ , setting $x=0$ leaves $c$ , because $(32)^0=1$ . Once you know that $x=0$ gives the minimum, define the given function in Desmos and evaluate $f(0)$ .
Hint 3
Write $\frac18$ as a power of $2$ . Then multiply its exponent by the entire exponent $4-\frac53x$ .

Step-by-step

Find the minimum, then rewrite with one base
Step 1
Locate the minimum
The base $\frac18$ is between $0$ and $1$ , so a smaller exponent gives a larger output. As $x$ increases, $4-\frac53x$ decreases, so $f(x)$ increases. On $x\ge0$ , the minimum occurs at the smallest allowed input, $x=0$ . Don't call this decay because the base is below $1$ ; the exponent is falling too.
Step 2
Evaluate the minimum in Desmos
Type the given function, then type $f(0)$ . Desmos shows $0.000244140625$ ; its fraction button gives $\frac1{4096}$ . Because $x=0$ gives the minimum, $c=f(0)=\frac1{4096}$ .
Step 3
Use a common base
Since $\frac18=2^{-3}$ , rewrite the original base:
$f(x)=\left(2^{-3}\right)^{4-\frac53x}.$
Step 4
Multiply the exponents
A power raised to a power multiplies its exponents, so the $-3$ applies to the whole exponent:
$f(x)=2^{-3\left(4-\frac53x\right)}.$
Step 5
Distribute through the exponent
Multiply $-3$ by both terms inside the parentheses:
$f(x)=2^{-12+5x}.$
Step 6
Separate the constant power
When powers with the same base multiply, their exponents add. So split the sum in the exponent:
$f(x)=2^{-12}2^{5x}.$
Step 7
Put x on the outside
Use the power-of-a-power rule in reverse so the exponent is $x$ alone:
$f(x)=2^{-12}(2^5)^x.$
Step 8
Match the requested base
Evaluate $2^5$ to get the base $32$ :
$f(x)=2^{-12}(32)^x.$
Step 9
Write the minimum as the coefficient
At $x=0$ , the last form gives $f(0)=2^{-12}$ . Desmos found that same value to be $\frac1{4096}$ , so replace the coefficient, not the base: $f(x)=\frac{1}{4096}(32)^x$ . Its coefficient is the minimum for $x\ge0$ . Choice A.

Question 14·200 Super-Hard SAT Math Questions·Advanced Math

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