Question

What is the simplest radical form of the expression?​

Answer 1

`4x^3y^2\sqrt[3]{x}`

Explanation:

Applying the exponent rule
`\left(a\cdot \:b\right)^n=a^nb^n`

`(8x^5y^3)^(2)/(3)` =
`8^{(2)/(3)}\left(x^5\right)^{(2)/(3)}\left(y^3\right)^{(2)/(3)}` =
`8^{(2)/(3)}\left(x^5\right)^{(2)/(3)}\left(y^3\right)^{(2)/(3)}`

`8^(2)/(3) = \sqrt[3]{8^2} = \sqrt[3]{64} = 4`

`(x^5})^(2)/(3) = x^(5.2)/(3) =x^(10)/(3)`

`(y^3)^(2)/(3) = y^(3.2)/(3) = y^2`

So
`(8x^5y^3)^(2)/(3) = 4x^(10)/(3)y^2`

We can eliminate the first and third choices since the coefficient is not 4

If we look at the other two choices 2 and 4

`\sqrt[3]{xy^2} = \sqrt[3]{x} \sqrt[3]{y^2}` and this cannot result in a
`y^2` term

Therefore the answer is
`4x^3y^2\sqrt[3]{x}`

We can verify this by converting the cube root to an exponent

`\sqrt[3]{x} = x^(1)/(3)` So we get

`4x^3y^2x^(1)/(3)`

Collecting like terms we get

`4x^(3)x^{(1)/(3)}y^(2)`

Noting that
`x^ax^b = x^(a+b)`

`x^(3)x^{(1)/(3)} = x^{3`
`x^{3+(1)/(3)}=x^(10)/(3)`

So the expression evaluates to

`4x^(10)/(3)y^2`

which fits our simplified value