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Advanced Math / Equivalent expressions Difficulty: Hard

If $4 Superscript 8 c Baseline equals RootIndex 3 StartRoot 4 Superscript 7 Baseline EndRoot$ , what is the value of $c$ ?

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Explanation

The correct answer is $\frac{7}{24}$ . An expression of the form $\sqrt[n]{a^{m}}$ , where $m$ and $n$ are integers greater than $1$ and $a greater than or equals 0$ , is equivalent to $a^{\frac{m}{n}}$ . Therefore, the expression on the right-hand side of the given equation, $RootIndex 3 StartRoot 4 Superscript 7 Baseline EndRoot$ , is equivalent to $4 Superscript seven thirds$ . Thus, $4 Superscript 8 c Baseline equals 4 Superscript seven thirds$ . It follows that $8 c equals seven thirds$ . Dividing both sides of this equation by $8$ yields $c = \frac{7}{24}$ . Note that 7/24, .2916, .2917, 0.219, and 0.292 are examples of ways to enter a correct answer.