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

Please explain me how it says $\sqrt[3]{x} \sqrt[3]{x} = {x}^{\frac{2}{3}}$

Mar 2, 2016

$\sqrt[3]{x} - \frac{1}{\sqrt[3]{x}}$

Take out the $L C D : \sqrt[3]{x}$

$\rightarrow \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x}} - \frac{1}{\sqrt[3]{x}}$

Make their denominators same

$\rightarrow \frac{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) - 1}{\sqrt[3]{x}}$

$\sqrt[3]{x} \cdot \sqrt[3]{x} = \sqrt[3]{x \cdot x} = \sqrt[3]{{x}^{2}} = {x}^{\frac{2}{3}}$

$\Rightarrow = \frac{{x}^{\frac{2}{3}} - 1}{\sqrt[3]{x}}$

Mar 2, 2016

$\textcolor{b l u e}{\text{Explaining the connection between "root(3)(x)root(3)(x)" and } {x}^{\frac{2}{3}}}$

Explanation:

$\textcolor{b l u e}{\text{Point 1}}$

Look at these alternative ways of writing roots

$\sqrt{x} \text{ is the same as } {x}^{\frac{1}{2}}$

$\sqrt[3]{x} \text{ is the same as } {x}^{\frac{1}{3}}$

$\sqrt[4]{x} \text{ is the same as } {x}^{\frac{1}{4}}$

So for any number $n \text{ "root(n)(x) " is the same as } {x}^{\frac{1}{n}}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Point 2}}$

Just picking a number at random I chose 3

Another way (not normally done) of writing 3 is ${3}^{1}$

When you have $3 \times 3 \text{ it can be written as } {3}^{2}$

In the same way $3 \times 3 \times 3 \text{ can be written as } {3}^{3}$

In the same way $3 \times 3 \times 3 \times 3 \text{ can be written as } {3}^{4}$

Notice that $3 \times 3 = {3}^{1} \times {3}^{1} = {3}^{1 + 1} = {3}^{2}$

Notice that $3 \times 3 \times 3 = {3}^{1} \times {3}^{1} \times {3}^{1} = {3}^{1 + 1 + 1} = {3}^{3}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Point 3}}$

Given that a way of writing square root of 3 is $\sqrt{3} \text{ is } {3}^{\frac{1}{2}}$

Compare what happens in each of the following two rows

${3}^{1} \times {3}^{1} \times {3}^{1} = {3}^{1 + 1 + 1} = {3}^{3}$

${3}^{\frac{1}{2}} \times {3}^{\frac{1}{2}} \times {3}^{\frac{1}{2}} = {3}^{\frac{1}{2} + \frac{1}{2} + \frac{1}{2}} = {3}^{\frac{3}{2}}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Point 4}}$

$\textcolor{b r o w n}{\text{You asked about } \sqrt[3]{x} \sqrt[3]{x} = {x}^{\frac{2}{3}}}$

From above we know that $\sqrt[3]{x} \text{ is the same as } {x}^{\frac{1}{3}}$

But we have $\sqrt[3]{x} \sqrt[3]{x}$

This is the same as ${x}^{\frac{1}{3}} \times {x}^{\frac{1}{3}} = {x}^{\frac{1}{3} + \frac{1}{3}} = {x}^{\frac{2}{3}}$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Point 5}}$

Backtrack for a moment and again think about

${x}^{\frac{1}{3}} \times {x}^{\frac{1}{3}}$

Like in $3 \times 3 = {3}^{2}$

${x}^{\frac{1}{3}} \times {x}^{\frac{1}{3}} = {\left({x}^{\frac{1}{3}}\right)}^{2}$

and ${x}^{\frac{1}{3}} \times {x}^{\frac{1}{3}} = {x}^{\frac{1}{3} + \frac{1}{3}} = {x}^{\frac{2}{3}}$

Then ${\left({x}^{\frac{\textcolor{m a \ge n t a}{1}}{3}}\right)}^{\textcolor{g r e e n}{2}} = {x}^{\frac{\textcolor{m a \ge n t a}{1} \times \textcolor{g r e e n}{2}}{3}} = {x}^{\frac{2}{3}}$

Turning this back the other way

${x}^{\frac{2}{3}} = \sqrt[3]{{x}^{2}}$

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!!