Question

Simplify 2+^3 ÷ 2- ^3

Answer 1

Simplifying a fraction

We want to simplify the following expression:

`\frac{2+\sqrt[]{3}}{2-\sqrt[]{3}}`

This means that we want to "remove" the denominator".

STEP 1

If we observe the denominator:

`(2-\sqrt[]{3})`

If we multiply it by

2 + √3, then

`\begin{gathered} (2-\sqrt[]{3})(2+\sqrt[]{3}) \\ =4-\sqrt[]{3}^2=4-3=1 \end{gathered}`

STEP 2

We know that if we multiply both sides of a fraction by the same number or expression, the fraction will remain the same, then we multiply both sides by 2 + √3:

`\frac{2+\sqrt[]{3}}{2-\sqrt[]{3}}=\frac{(2+\sqrt[]{3})(2+\sqrt[]{3})}{(2-\sqrt[]{3})(2+\sqrt[]{3})}`

For the denominator, as we analyzed before

`(2-\sqrt[]{3})(2+\sqrt[]{3})=1`

For the denominator:

`(2+\sqrt[]{3})(2+\sqrt[]{3})=(2+\sqrt[]{3})^2`

Then,

`\frac{2+\sqrt[]{3}}{2-\sqrt[]{3}}=\frac{(2+\sqrt[]{3})(2+\sqrt[]{3})}{(2-\sqrt[]{3})(2+\sqrt[]{3})}=\frac{(2+\sqrt[]{3})^2}{1}=(2+\sqrt[]{3})^2`

STEP 3

Now, we can simplify the result:

`\begin{gathered} (2+\sqrt[]{3})^2=(2+\sqrt[]{3})(2+\sqrt[]{3}) \\ =2^2+2\sqrt[]{3}+(\sqrt[]{3})^2+2\sqrt[]{3} \\ =4+4\sqrt[]{3}+3 \\ =7+4\sqrt[]{3} \end{gathered}`Answer: 7+4√3