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Advanced Math / Equivalent expressions Difficulty: Hard

The expression $the fraction with numerator x to the power negative 2, end power, times y to the power one-half, and denominator x to the power one-third, end power, times y to the power negative 1, end fraction$ , where $x is greater than 1$ and $y is greater than 1$ , is equivalent to which of the following?

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Explanation

Choice D is correct. For $x is greater than 1$ and $y is greater than 1$ , $x to the one third power$ and $y to the one half power$ are equivalent to $the cube root of x$ and $the square root of y$ , respectively. Also, $x to the negative 2 power$ and $y to the negative 1 power$ are equivalent to $the fraction 1 over x squared, end fraction$ and $1 over y$ , respectively. Therefore, the given expression can be rewritten as $the fraction with numerator, y times the square root of y, and denominator x squared, times the cube root of x, end fraction$ .

Choices A, B, and C are incorrect because these choices are not equivalent to the given expression for $x is greater than 1$ and $y is greater than 1$ .

For example, for $x equals 2$ and $y equals 2$ , the value of the given expression is $2 to the negative five sixths power$ ; the values of the choices, however, are $2 to the negative one third power$ , $2 to the five sixths power$ , and 1, respectively.