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Advanced Math / Equivalent expressions Difficulty: Medium

Which expression is equivalent to $StartFraction h Superscript 15 Baseline q Superscript 7 Baseline Over h Superscript 5 Baseline q Superscript 21 Baseline EndFraction$ , where $h greater than 0$ and $q greater than 0$ ?

$StartFraction h Superscript 10 Baseline Over q Superscript 14 Baseline EndFraction$

$\frac{h^{3}}{q^{3}}$

$h Superscript 10 Baseline q Superscript 14$

$h^{3} q^{3}$

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Explanation

Choice A is correct. For positive values of $a$ , $\frac{a^{m}}{a^{n}} = a^{(m - n)}$ , where $m$ and $n$ are integers. Since it's given that $h greater than 0$ and $q greater than 0$ , this property can be applied to rewrite the given expression as $(h^{(15 - 5)}) (q^{(7 - 21)})$ , which is equivalent to $h Superscript 10 Baseline q Superscript negative 14$ . For positive values of $a$ , $a^{- n} = \frac{1}{a^{n}}$ . This property can be applied to rewrite the expression $h Superscript 10 Baseline q Superscript negative 14$ as $left parenthesis h Superscript 10 Baseline right parenthesis left parenthesis StartFraction 1 Over q Superscript 14 Baseline EndFraction right parenthesis$ , which is equivalent to $StartFraction h Superscript 10 Baseline Over q Superscript 14 Baseline EndFraction$ .

Choice B 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.