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Advanced Math / Equivalent expressions Difficulty: Medium

Which expression is equivalent to $a Superscript StartFraction 11 Over 12 EndFraction$ , where $a greater than 0$ ?

$RootIndex 12 StartRoot a Superscript 132 Baseline EndRoot$

$RootIndex 144 StartRoot a Superscript 132 Baseline EndRoot$

$RootIndex 121 StartRoot a Superscript 132 Baseline EndRoot$

$RootIndex 11 StartRoot a Superscript 132 Baseline EndRoot$

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Explanation

Choice B is correct. Since $\frac{12}{12} = 1$ , multiplying the exponent of the given expression by $StartFraction 12 Over 12 EndFraction$ yields an equivalent expression: $a^{(\frac{11}{12}) (\frac{12}{12})} = a^{(\frac{132}{144})}$ . Since $\frac{132}{144} = 132 (\frac{1}{144})$ , the expression $a Superscript StartFraction 132 Over 144 EndFraction$ can be rewritten as $a Superscript left parenthesis 132 right parenthesis left parenthesis StartFraction 1 Over 144 EndFraction right parenthesis$ . Applying properties of exponents, this expression can be rewritten as $left parenthesis a Superscript 132 Baseline right parenthesis Superscript StartFraction 1 Over 144 EndFraction$ . An expression of the form $left parenthesis m right parenthesis Superscript StartFraction 1 Over k EndFraction$ , where $m greater than 0$ and $k greater than 0$ , is equivalent to $\sqrt[k]{m}$ . Therefore, $left parenthesis a Superscript 132 Baseline right parenthesis Superscript StartFraction 1 Over 144 EndFraction$ is equivalent to $RootIndex 144 StartRoot a Superscript 132 Baseline EndRoot$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.