Question

Please help, 50 points

Convert
`5^{(2)/(3) } x^{(4)/(3) } yz^(2)` to radical form

Answer 1

`5\cdot √(5)\cdot \sqrt[3]{x^4}\cdot y\cdot z^2` =
`5\cdot √(5)\cdot x\sqrt[3]{x}\cdot y\cdot z^2`

Explanation:

Let's take the individual terms and convert them to radical form

`5^{(3)/(2)} = (5^3)}^{(1)/(2)} = 125^{(1)/(2)`

A term raised to the power 1/2 is nothing but the square root of that term

`125^{(1)/(2)} = √(125) = √(25\cdot 5) = √(25) √(5) = 5 √(5)`

`x^{(4)/(3)} = (x^4)^{(1)/(3)} = \sqrt[3]{x^4}` since any term raised to 1/3 is the cube root of that term

We can simplify this further by noting
`x^4 = x^3 \cdot x`'

So
`\sqrt[3]{x^4} = \sqrt[3]{x^3} x = x\sqrt[3]{x}`

The other exponents are integers

Putting them all together we get

`5^{(3)/(2) }x^{(4)/(3)}yz^2 = 5\cdot √(5)\cdot x\sqrt[3]{x}\cdot y\cdot z^2`