Question 148114
*[Tex \LARGE \frac{\left(x^{\frac{3}{2}}y^{-3}\right)^{\frac{1}{3}}}{\left(x^{\frac{1}{2}}y^{-6}\right)^{-1}}] Start with the given expression.



Distribute the outer exponents to the inner exponents. Remember, {{{(x^(y))^z=x^(y*z)}}}. So in our case the numerator goes from  *[Tex \LARGE \left(x^{\frac{3}{2}}y^{-3}\right)^{\frac{1}{3}}] to  *[Tex \LARGE x^{\left(\frac{3}{2}\right)\left(\frac{1}{3}\right)}y^{\left(-3\right)\left(\frac{1}{3}\right)}]and the denominator goes from  *[Tex \LARGE \left(x^{\frac{1}{2}}y^{-6}\right)^{-1}] to *[Tex \LARGE x^{\left(\frac{1}{2}\right)\left(-1\right)}y^{\left(-6\right)\left(-1\right)}]



*[Tex \LARGE \frac{x^{\left(\frac{3}{2}\right)\left(\frac{1}{3}\right)}y^{\left(-3\right)\left(\frac{1}{3}\right)}}{x^{\left(\frac{1}{2}\right)\left(-1\right)}y^{\left(-6\right)\left(-1\right)}}]  Use the technique described above to distribute the exponents.



*[Tex \LARGE \frac{x^{\frac{3}{6}}y^{-\frac{3}{3}}}{x^{-\frac{1}{2}}y^{6}}] Multiply the exponents.



*[Tex \LARGE \frac{x^{\frac{1}{2}}y^{-1}}{x^{-\frac{1}{2}}y^{6}}] Reduce. 




Now when we divide monomials, we simply subtract the exponents. So for example {{{x^5/x^2=x^(5-2)=x^3}}}


Let's apply this technique to the problem:


*[Tex \LARGE x^{\frac{1}{2}-\left(-\frac{1}{2}\right)}y^{-1-6}] Subtract the exponents.


*[Tex \LARGE x^{\frac{1}{2}+\frac{1}{2}}y^{-1-6}] Rewrite *[Tex \LARGE \frac{1}{2}-\left(-\frac{1}{2}\right)] as *[Tex \LARGE \frac{1}{2}+\frac{1}{2}]



*[Tex \LARGE x^{\frac{2}{2}}y^{-7}] Combine the exponents.



*[Tex \LARGE x^{1}y^{-7}] Reduce.



*[Tex \LARGE xy^{-7}] Simplify



*[Tex \LARGE \frac{x}{y^{7}}] Now rewrite {{{y^(-7)}}} as {{{1/y^7}}}




So *[Tex \LARGE \frac{\left(x^{\frac{3}{2}}y^{-3}\right)^{\frac{1}{3}}}{\left(x^{\frac{1}{2}}y^{-6}\right)^{-1}}] simplifies to *[Tex \LARGE \frac{x}{y^{7}}] 


In other words, *[Tex \LARGE \frac{\left(x^{\frac{3}{2}}y^{-3}\right)^{\frac{1}{3}}}{\left(x^{\frac{1}{2}}y^{-6}\right)^{-1}}=\frac{x}{y^{7}}] where every variable is positive and {{{x<>0}}} or {{{y<>0}}}