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Advanced Math / Equivalent expressions Difficulty: Hard

$RootIndex 5 StartRoot 70 n EndRoot left parenthesis RootIndex 6 StartRoot 70 n EndRoot right parenthesis squared$

For what value of $x$ is the given expression equivalent to $left parenthesis 70 n right parenthesis Superscript 30 x$ , where $n greater than 1$ ?

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Explanation

The correct answer is $StartFraction 4 Over 225 EndFraction$ . An expression of the form $\sqrt[k]{a}$ , where $k$ is an integer greater than $1$ and $a greater than or equals 0$ , is equivalent to $a Superscript StartFraction 1 Over k EndFraction$ . Therefore, the given expression, where $n greater than 1$ , is equivalent to $left parenthesis 70 n right parenthesis Superscript one fifth Baseline left parenthesis left parenthesis 70 n right parenthesis Superscript one sixth Baseline right parenthesis squared$ . Applying properties of exponents, this expression can be rewritten as $left parenthesis 70 n right parenthesis Superscript one fifth Baseline left parenthesis 70 n right parenthesis Superscript one sixth dot 2$ , or $left parenthesis 70 n right parenthesis Superscript one fifth Baseline left parenthesis 70 n right parenthesis Superscript one third$ , which can be rewritten as $left parenthesis 70 n right parenthesis Superscript one fifth plus one third$ , or $left parenthesis 70 n right parenthesis Superscript eight fifteenths$ . It's given that the expression $RootIndex 5 StartRoot 70 n EndRoot left parenthesis RootIndex 6 StartRoot 70 n EndRoot right parenthesis squared$ is equivalent to $left parenthesis 70 n right parenthesis Superscript 30 x$ , where $n greater than 1$ . It follows that $left parenthesis 70 n right parenthesis Superscript eight fifteenths$ is equivalent to $left parenthesis 70 n right parenthesis Superscript 30 x$ . Therefore, $eight fifteenths equals 30 x$ . Dividing both sides of this equation by $30$ yields $\frac{8}{450} = x$ , or $\frac{4}{225} = x$ . Thus, the value of $x$ for which the given expression is equivalent to $left parenthesis 70 n right parenthesis Superscript 30 x$ , where $n greater than 1$ , is $StartFraction 4 Over 225 EndFraction$ . Note that 4/225, .0177, .0178, 0.017, and 0.018 are examples of ways to enter a correct answer.