Question

Simplify and write in radical form:
((x^3 * y^-2) / xy)^-(1/5)

Answer 1

`\sqrt[5]{((y^3)/(x^2))}`

Explanation:

`((x^3* y^(-2))/(xy))^{(-1)/(5)}`

Now we will use the quotient rule of exponent which is given by

`(x^a)/(x^b)=a^(a-b)`

Using this property, we get

`({x^(3-1)* y^(-2-1)})^{(-1)/(5)}`

On simplifying, we get

`({x^(2)* y^(-3)})^{(-1)/(5)}`

Now, use the property,
`x^(-a)=(1)/(x^a)`

`((x^2)/(y^3))^{(-1)/(5)}`

Now, we can use the rule
`x^(1/a)=\sqrt[a]{x}`

`((y^3)/(x^2))^{(1)/(5)}\\\\=\sqrt[5]{((y^3)/(x^2))}`

Answer 2

The answer is
`\sqrt[5]{(x^(2))/(y^(3))}`


`( ( x^(3)* y^(-2) )/(xy))^{ (1)/(5) } \\ \\ ( x^(a) )/( x^(b) ) = x^(a-b) \\ \\ ( ( x^(3)* y^(-2) )/(xy))^{ (1)/(5) } = ( x^(3-1)y^(-2-1) ) ^{ (1)/(5) } = ( x^(2)y^(-3) ) ^{ (1)/(5) } \\ \\ x^(-a) = (1)/( x^(a) ) \\ \\ ( x^(2)y^(-3) ) ^(2)= ((x^(2))/(y^(3)) )^{ (1)/(5) } \\ \\ x^{ (a)/(b) } = \sqrt[b]{ x^(a) } \\ \\ ((x^(2))/(y^(3)) )^{ (1)/(5) } = \sqrt[5]{(x^(2))/(y^(3))}`