Two numbers and are each greater than zero. Suppose and Which choice is equal to ?

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Super-Hard SAT Math Question 70 | Free SAT Prep | Aniko

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Question 70·200 Super-Hard SAT Math Questions·Advanced Math

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Two numbers $r$ and $s$ are each greater than zero. Suppose

\sqrt[4]{r}=\sqrt[7]{s}

and

r^2s^3=2^{29}.

Which choice is equal to $\sqrt[14]{r^5s^4}$ ?

When two different radicals are equal, introduce a shared variable (like $k$ ) for their common value and rewrite each original quantity as a power of $k$ . Then use the additional equation to solve for $k$ (here it becomes a simple power equation), and finally rewrite the target expression in terms of $k$ and simplify exponents carefully, applying the outer root as multiplication by a fraction.

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Create a single shared base

Since $\sqrt[4]{r}=\sqrt[7]{s}$ and both numbers are positive, set both equal to the same variable (for example, $k$ ).

Rewrite $r$ and $s$ as powers of that base

If $k=\sqrt[4]{r}$ , then $r$ is a power of $k$ . Do the same for $s$ using $k=\sqrt[7]{s}$ .

Be careful with the 14th root

Rewrite $\sqrt[14]{r^5s^4}$ using fractional exponents and combine exponents before substituting the value of $k$ .

Desmos Guide

Find the common base $k$ from $k^{29}=2^{29}$

Graph $y=(x/2)^{29}$ and $y=1$ . The positive intersection’s $x$ -value is $k$ .

Compute the transformed expression in terms of $k$

Enter $y=x^{24/7}$ . Then evaluate this at the $x$ -value found for $k$ (for example, by creating a point $(k, k^{24/7})$ using the intersection value).

Match to an answer choice

Compare the expression you evaluated (written as a power of 2) to the four choices to select the match.

Step-by-step Explanation

Rewrite $r$ and $s$ using a common base

Let $k=\sqrt[4]{r}=\sqrt[7]{s}$ .

Then

r=k^4 \quad\text{and}\quad s=k^7.

Use $r^2s^3=2^{29}$ to find $k$

Substitute $r=k^4$ and $s=k^7$ into $r^2s^3$ :

\begin{aligned} r^2s^3 &= (k^4)^2(k^7)^3 \\ &= k^8\cdot k^{21} \\ &= k^{29}. \end{aligned}

So $k^{29}=2^{29}$ . Since $k>0$ , it follows that $k=2$ .

Rewrite the target expression in terms of $k$

\begin{aligned} \sqrt[14]{r^5s^4} &= (r^5s^4)^{1/14} \\ &= \big((k^4)^5 (k^7)^4\big)^{1/14} \\ &= \big(k^{20}k^{28}\big)^{1/14} \\ &= \left(k^{48}\right)^{1/14} \\ &= k^{48/14} \\ &= k^{24/7}. \end{aligned}

Substitute $k=2$

With $k=2$ , the expression equals $2^{24/7}$ .

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Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation