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# Perform the indicated operations, expressing answers in simplest form with rationalized denominators.$\frac{\sqrt{T^{4}-V^{4}}}{\sqrt{V^{-2}-T^{-2}}}$

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For the following expression:

$\dfrac{\sqrt{T^4-V^4}}{\sqrt{V^{-2}-T^{-2}}}$

To rationalize the denominator, multiply the numerator and the denominator by $\sqrt{V^{-2}-T^{-2}}$ as follows:

$\ \ \ \ \ \ \ \ =\dfrac{\sqrt{T^4-V^4}}{\sqrt{V^{-2}-T^{-2}}} \times\dfrac{\sqrt{V^{-2}-T^{-2}}}{\sqrt{V^{-2}-T^{-2}}}$

Apply this radical rule $\sqrt{a}\sqrt{a}=a$ as follows:

$\ =\dfrac{\sqrt{T^4-V^4}\sqrt{V^{-2}-T^{-2}}}{V^{-2}-T^{-2}}$

Simplify $V^{-2}-T^{-2}:\ \dfrac{1}{V^2}-\dfrac{1}{T^2}$ using this rule $a^{-n}=\dfrac{1}{a^n}$ as follows:

$=\dfrac{\sqrt{T^4-V^4}\sqrt{ \frac{1}{V^2}-\frac{1}{T^2}}}{\frac{1}{V^2}-\frac{1}{T^2}}$

Combine the fractions $\dfrac{1}{V^2}-\dfrac{1}{T^2}=\dfrac{T2-V^2}{V^2T^2}$ as follows:

$\ \ =\dfrac{\sqrt{T^4-V^4}\sqrt{\dfrac{T^2-V^2}{T^2V^2}}}{\dfrac{T2-V^2}{V^2T^2}}$

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