Question

simplify 2√3 upon 3 minus √3 upon 6

Answer 1

We have,

2√3
_____
3 - √3
_____
6

=> 2√3 x 6
________
3 - √ 3

=> 12 √ 3 3 + √ 3
______ x _________
3 - √ 3 3 + √ 3

=> 36√3 + 36
___________
9 - 3

=> 36 ( √3 + 1 )
___________
6

=> 6 √3 + 6

Answer 2

Hey there!

The following numbers are given and is told to divide it two times (not necessarily). Here we're asked to simplify the whole values and bring out a final answer.

Now, by applying the fraction rule that is,

$\bf{\frac{a}{\frac{b}{c}}}$

$\bf{= \frac{a \times c}{b}}$

Now,

$\bf{= \frac{2 \times 6 \sqrt{3}}{3 - \sqrt{3}}}$

By applying and multiplying the following numbers 2 × 6 = 12.

$\bf{= \frac{12 \sqrt{3}}{3 - \sqrt{3}}}$

Factoring out $\bf{3 - \sqrt{3}}$, 3 = $\bf{\sqrt{3} \times \sqrt{3}}$.

$\bf{= \frac{12 \sqrt{3}}{\sqrt{3} \sqrt{3} - \sqrt{3}}}$

Factor out the common term $\bf{\sqrt{3}}$.

$\bf{= \frac{12 \sqrt{3} }{\sqrt{3}(\sqrt{3} - 1)}}$

Cancel out the common term of factor $\bf{\sqrt{3}}$.

$\bf{= \frac{12}{\sqrt{3} - 1}}$

Rationalising the whole value;

Multiplying by a conjugated term that is,

$\bf{\frac{\sqrt{3} + 1}{\sqrt{3} + 1}}$

Therefore,

$\bf{= \frac{12(\sqrt{3} + 1)}{(\sqrt{3} - 1) \times (\sqrt{3} + 1)}}$

Apply the rule of Difference of two squares formula that is,

$\bf{(a - b) \times (a + b) = {a}^{2} - {b}^{2}}$

$\bf{Here, \: \: \: a = \sqrt{3}, \: \: \: b = 1}$

Therefore,

$\bf{= \frac{12(\sqrt{3} + 1) }{{(\sqrt{3})}^{2} - {1}^{2}}}$

Apply this rule $\bf{1^a = 1}$.

Here, $\bf{1^2 = 1}$.

Now,

$\bf{= \frac{12(\sqrt{3} + 1)}{{(\sqrt{3})}^{2} - 1}}$

Applying the rule of square roots to half squares that is,

$\bf{\sqrt{a} = {a}^{\frac{1}{2}}}$

Therefore,

$\bf{= \frac{12(\sqrt{3} + 1)}{{({3}^{\frac{1}{2}})}^{2} - 1}}$

Apply the rule of exponential forms that is,

$\bf{({{a}^{b}})^{c} = {a}^{b \times c}}$

Here,

$\bf{= \frac{12(\sqrt{3} + 1)}{{3}^{\frac{1}{2} \times 2} - 1}}$

Multiply the fractions that is,

$\bf{a \times \frac{b}{c} = \frac{a \times b}{c}}$

Here,

$\bf{= \frac{12(\sqrt{3} + 1)}{3 - 1}}$

Subtract the numbers,

$\bf{= \frac{12(\sqrt{3} + 1)}{2}}$

Now, divide the numbers to obtain the final value,

$\boxed{\huge{\bf{\underline{= 6(1 + \sqrt{3})}}}}$

Which is the final solution for this type of query.

Hope this helps and clears your doubts in a detailed way !