Seelakant6vr

2022-02-02

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

Step 1 $\sqrt[3]{x}-\frac{1}{\sqrt[3]{x}}$ Take out the $LCD:\sqrt[3]{x}$ $\to \frac{\sqrt[3]{x}×\sqrt[3]{x}}{\sqrt[3]{x}}-\frac{1}{\sqrt[3]{x}}$ Make their denominators same $\to \frac{\left(\sqrt[3]{x}×\sqrt[3]{x}\right)-1}{\sqrt[3]{x}}$ $\sqrt[3]{x}×\sqrt[3]{x}=\sqrt[3]{x×x}=\sqrt[3]{{x}^{2}}={x}^{\frac{2}{3}}$ $⇒=\frac{{x}^{\frac{2}{3}}-1}{\sqrt[3]{x}}$

Amiya Wolf

Step 1 Look at these alternative ways of writing roots $\sqrt{x}$ is the same as ${x}^{\frac{1}{2}}$ $\sqrt[3]{x}$ is the same as ${x}^{\frac{1}{3}}$ $\sqrt[4]{x}$ is the same as ${x}^{\frac{1}{4}}$ So for any number n $\sqrt[n]{x}$ is the same as ${x}^{\frac{1}{n}}$ Step 2 Just picking a number at random I chose 3 Another way (not normally done) of writing 3 is ${3}^{1}$ When you have $3×3$ it can be written as ${3}^{2}$ In the same way $3×3×3$ can be written as ${3}^{3}$ In the same way $3×3×3×3$ can be written as ${3}^{4}$ Notice that $3×3={3}^{1}×{3}^{1}={3}^{1+1}={3}^{2}$ Notice that $3×3×3={3}^{1}×{3}^{1}×{3}^{1}={3}^{\left(}1+1+1\right)={3}^{3}$ Step 3 Given that a way of writing square root of 3 is $\sqrt{3}$ is ${3}^{\frac{1}{2}}$ Compare what happens in each of the following two rows ${3}^{1}×{3}^{1}×{3}^{1}={3}^{1+1+1}={3}^{3}$ ${3}^{\frac{1}{2}}×{3}^{\frac{1}{2}}×{3}^{\frac{1}{2}}={3}^{\frac{1}{2}+\frac{1}{2}+\frac{1}{2}}={3}^{\frac{3}{2}}$ Step 4 You asked about $\sqrt[3]{x}\sqrt[3]{x}={x}^{\frac{2}{3}}$ From above we know that $\sqrt[3]{x}$ is the same as ${x}^{\frac{1}{3}}$ But we have $\sqrt[3]{x}\sqrt[3]{x}$ This is the same as ${x}^{\frac{1}{3}}×{x}^{\frac{1}{3}}={x}^{\frac{1}{3}+\frac{1}{3}}={x}^{\frac{2}{3}}$ Step 5 Backtrack for a moment and again think about ${x}^{\frac{1}{3}}×{x}^{\frac{1}{3}}$ Like in $3×3={3}^{2}$ ${x}^{\frac{1}{3}}×{x}^{\frac{1}{3}}=\left({x}^{\frac{1}{3}}{\right)}^{2}$ and ${x}^{\frac{1}{3}}×{x}^{\frac{1}{3}}={x}^{\frac{1}{3}+\frac{1}{3}}={x}^{\frac{2}{3}}$ Then $\left({x}^{\frac{1}{3}}{\right)}^{2}={x}^{\frac{1×2}{3}}={x}^{\frac{2}{3}}$ 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.