Question 169·200 Super-Hard SAT Math Questions·Advanced Math

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Two positive numbers $r$ and $s$ , with $r\ne 1$ , satisfy $\sqrt[3]{r}=\sqrt[5]{s}$ . Which choice gives the value of $x$ for which $r^{10x-3}\,s^{2-x}$ is equal to $s^{4-2x}\,r^{x+1}$ ?

Equal roots of two positive numbers signal an exponential equation that can be rewritten with one common base. Name the shared root, express each number as a power of it, and match exponents. A Desmos regression can then solve the resulting equation. Keep the constant terms inside each exponent when you rewrite the powers.

Hints

Hint 1
The number on a root tells you which power undoes it: cubing a cube root gives the original number. Call the shared root $t$ . What powers of $t$ give $r$ and $s$ ?
Hint 2
Use the power-of-a-power rule: $(t^a)^b=t^{ab}$ . When you rewrite each factor, multiply by the whole exponent, including its constant term.
Hint 3
Equal powers of the same positive base other than $1$ have equal exponents. How does $r\ne 1$ ensure that the shared root isn't $1$ ?

Step-by-step

Rewrite with one base, then solve
Step 1
Find a common base
Call the shared root $t$ : $t=\sqrt[3]{r}=\sqrt[5]{s}$ . A root is undone by its matching power, so $r=t^3$ and $s=t^5$ . Since $r\ne1$ , $t\ne1$ too.
Step 2
Replace both numbers with powers of the shared root
Substitute $r=t^3$ and $s=t^5$ into the equality the question asks about:
$(t^3)^{10x-3}(t^5)^{2-x}=(t^5)^{4-2x}(t^3)^{x+1}.$
Keep each entire exponent attached to its factor, especially the $-3$ and $+1$ .
Step 3
Write one power on each side
For a power of a power, multiply exponents; when multiplying powers of the same base, add exponents. Apply both rules to the whole product on each side:
$t^{3(10x-3)+5(2-x)}=t^{5(4-2x)+3(x+1)}.$
Step 4
Match the exponents
The base $t$ is positive and not 1. Equal powers of that base must have equal exponents, so:
$3(10x-3)+5(2-x)=5(4-2x)+3(x+1).$
If $t$ were $1$ , every power would equal $1$ , and matching exponents wouldn't work. Match the bases first, then match the exponents.
Step 5
Solve for the requested value
Type $3(10x_1-3)+5(2-x_1)\sim5(4-2x_1)+3(x_1+1)$ in Desmos. Use $x_1$ for the unknown and $\sim$ to ask Desmos to find it. Under PARAMETERS, Desmos shows $x_1=0.6875$ . Type $x_1$ on the next line and use its fraction button to see $\frac{11}{16}$ . So $x=\frac{11}{16}$ makes the original products equal. Choice C.

Question 169·200 Super-Hard SAT Math Questions·Advanced Math

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