Question

What is `root(3)x-1/(root(3)x)?`

Please explain me how it says `root(3)xroot(3)x=x^(2/3)`

Answer 1

`root(3)x-1/(root(3)x)`

Take out the `LCD:root(3)x`

`rarr(root(3)x*root(3)x)/root(3)x-1/(root(3)x)`

Make their denominators same

`rarr((root(3)x*root(3)x)-1)/(root(3)x)`

`root(3)x*root(3)x=root(3)(x*x)=root(3)(x^2)=x^(2/3)`

`rArr=(x^(2/3)-1)/root(3)(x)`

Answer 2

`color(blue)("Explaining the connection between "root(3)(x)root(3)(x)" and "x^(2/3))`

Explanation:

`color(blue)("Point 1")`

Look at these alternative ways of writing roots

`sqrt(x)" is the same as " x^(1/2)`

`root(3)(x)" is the same as "x^(1/3)`

`root(4)(x)" is the same as "x^(1/4)`

So for any number `n" "root(n)(x) " is the same as "x^(1/n)`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Point 2")`

Just picking a number at random I chose 3

Another way (not normally done) of writing 3 is `3^1`

When you have `3xx3" it can be written as " 3^2`

In the same way `3xx3xx3" can be written as "3^3`

In the same way `3xx3xx3xx3" can be written as "3^4`

Notice that `3xx3 = 3^1xx3^1 =3^(1+1) = 3^2`

Notice that `3xx3xx3=3^1xx3^1xx3^1=3^(1+1+1)=3^3`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Point 3")`

Given that a way of writing square root of 3 is `sqrt(3)" is "3^(1/2)`

Compare what happens in each of the following two rows

`3^1xx3^1xx3^1 = 3^(1+1+1) = 3^3`

`3^(1/2)xx3^(1/2)xx3^(1/2) =3^(1/2+1/2+1/2)=3^(3/2)`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Point 4")`

`color(brown)("You asked about "root(3)(x)root(3)(x)=x^(2/3))`

From above we know that `root(3)(x)" is the same as "x^(1/3)`

But we have `root(3)(x)root(3)(x)`

This is the same as `x^(1/3)xxx^(1/3) = x^(1/3+1/3) = x^(2/3)`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Point 5")`

Backtrack for a moment and again think about

`x^(1/3)xxx^(1/3)`

Like in `3xx3 = 3^2`

`x^(1/3)xxx^(1/3)=(x^(1/3))^2`

and `x^(1/3)xxx^(1/3)=x^(1/3+1/3)=x^(2/3)`

Then `(x^((color(magenta)(1))/3))^(color(green)(2)) = x^((color(magenta)(1)xxcolor(green)(2))/3) = x^(2/3)`

Turning this back the other way

`x^(2/3) = root(3)(x^2)`

Practise and a lot of it will fix this in your mind. It will seem confusing at first but as you practise more and more it will suddenly click!

Hope this helps!!