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The expression (x^(-2)y^(1/2))/(x^(1/3) ...

The expression `(x^(-2)y^(1/2))/(x^(1/3) y^(-1))` , where x gt 1 and y gt 1 , is equivalent to which of the following ?

A

`sqrty/(root3(x^2))`

B

`(ysqrty)/(root3(x^2))`

C

`(ysqrty)/(xsqrtx)`

D

`(ysqrty)/(x^2 root3(x))`

Text Solution

AI Generated Solution

The correct Answer is:
To simplify the expression \((x^{-2}y^{1/2})/(x^{1/3} y^{-1})\), we will follow these steps: ### Step 1: Rewrite the expression We start with the given expression: \[ \frac{x^{-2}y^{1/2}}{x^{1/3}y^{-1}} \] ### Step 2: Apply the properties of exponents Recall that \(a^{-b} = \frac{1}{a^b}\) and \(y^{-1} = \frac{1}{y}\). We can rewrite the expression as: \[ \frac{y^{1/2}}{y^{-1}} \cdot \frac{x^{-2}}{x^{1/3}} \] ### Step 3: Simplify the \(y\) terms Using the property of exponents \(y^{a}/y^{b} = y^{a-b}\): \[ y^{1/2 - (-1)} = y^{1/2 + 1} = y^{1/2 + 2/2} = y^{3/2} \] ### Step 4: Simplify the \(x\) terms Similarly, for the \(x\) terms: \[ x^{-2}/x^{1/3} = x^{-2 - 1/3} = x^{-2 - 3/3} = x^{-6/3 - 1/3} = x^{-7/3} \] ### Step 5: Combine the results Now we can combine the simplified \(x\) and \(y\) terms: \[ \frac{y^{3/2}}{x^{7/3}} = y^{3/2} \cdot x^{-7/3} \] ### Step 6: Final expression Thus, the expression simplifies to: \[ \frac{y^{3/2}}{x^{7/3}} \]
To simplify the expression \((x^{-2}y^{1/2})/(x^{1/3} y^{-1})\), we will follow these steps: ### Step 1: Rewrite the expression We start with the given expression: \[ \frac{x^{-2}y^{1/2}}{x^{1/3}y^{-1}} \] ...
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