Question 1·200 Super-Hard SAT Math Questions·Advanced Math

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In the given equations, $a$ and $b$ are constants, $a>1$ and $b>1$ . Which choice is the value of $x$ ?

\sqrt[8]{a^3}=\sqrt[6]{b^5}

a^{2x-1}=b^5

A pair of exponential equations with different bases becomes manageable when one equation contains a power from the other. Rewrite the roots as fractional powers, then raise both sides to match that shared power. Once the bases match, use a Desmos regression to solve for the exponent. The trap is equating exponents before the bases match.

Hints

Hint 1
An eighth root means a power of $1/8$ . For $a^3$ , multiply $3$ by $1/8$ to rewrite the root. How would you rewrite the sixth root of $b^5$ ?
Hint 2
The second equation contains $b^5$ . Raise both sides of the first equation to the sixth power so that $b^{5/6}$ becomes $b^5$ . What power of $a$ does that produce?
Hint 3
Equal powers of the same base greater than $1$ have equal exponents. Once you've replaced $b^5$ in the second equation, which two exponents can you set equal?

Step-by-step

Match the shared power
Step 1
Rewrite the roots as powers
A fractional exponent writes a root as a power: the root number goes in the denominator. Rewrite both sides of the first equation:
$a^{3/8}=b^{5/6}.$
Step 2
Make the power of $b$ match
The other equation contains $b^5$ , so raise both sides to the sixth power to produce $b^5$ :
$(a^{3/8})^6=(b^{5/6})^6.$
Multiply the exponents in each power of a power:
$a^{18/8}=b^{30/6}.$
Reduce the fractions:
$a^{9/4}=b^5.$
Step 3
Replace the shared power
The first equation now tells you what $b^5$ equals. Substitute $a^{9/4}$ for $b^5$ in the second equation:
$a^{2x-1}=a^{9/4}.$
Step 4
Set the exponents equal
Because $a>1$ , a larger exponent gives a larger power of $a$ . The powers now have the same base, so they can be equal only if their exponents match:
$2x-1=\frac94.$
Don't set $3/8$ equal to $5/6$ in the first equation: those powers have different bases.
Step 5
Solve for $x$ in Desmos
Type $2x_1-1\sim9/4$ . The $\sim$ tells Desmos to find $x_1$ rather than graph an equation in $x$ . Under PARAMETERS, it reports $x_1=1.625$ . Type $x_1$ on the next line and use its fraction button to see $13/8$ . So $x=\frac{13}{8}$ . Choice B.

Question 1·200 Super-Hard SAT Math Questions·Advanced Math

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