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

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

2 Answers
Mar 2, 2016

#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)#

Mar 2, 2016

Answer:

#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!!