\(\displaystyle \small \sqrt3(\sqrt4-\sqrt3)\)

\(\displaystyle \small \sqrt{12}-\sqrt9\)

\(\displaystyle \small \sqrt{12}=\sqrt4 \times \sqrt3=2\sqrt3\)

\(\displaystyle \small \sqrt9=3\)

\(\displaystyle \small \sqrt{12}-\sqrt9=2\sqrt3-3\)

FOIL with difference of squares.  The multiplying cancels the square roots on both terms.  

We can solve this by simplifying the radicals first: \(\displaystyle \sqrt{36} =6\)

Plugging this into the equation gives us: 

\(\displaystyle 6 \times3(6)=108\)

Note: the product of the radicals is the same as the radical of the product: 

\(\displaystyle \sqrt{2} \times \sqrt2} = \sqrt{2\times 2\)  which is \(\displaystyle \sqrt{4} =2\)

Once we understand this, we can plug it into the equation:

 \(\displaystyle 5\sqrt{2} \times4\sqrt{2} = 5\times4\times \sqrt{2\times 2}\)

\(\displaystyle 5\times4\times2=40\)

We can simplify the radicals:

\(\displaystyle \sqrt{32} =\sqrt{16\times 2} =4\sqrt{2}\)     and    \(\displaystyle \sqrt{16} =4\)

Plug in the simplifed radicals into the equation:

\(\displaystyle \frac{4\sqrt{2}}{3\times 4} = \frac{4\sqrt{2}}{12}\)

\(\displaystyle \frac{4 \sqrt2}{12}=\frac{\sqrt2}{3}\)

We can only simplify the radical in the numerator:  \(\displaystyle \sqrt{16} =4\)

Plugging in the simplifed radical into the equation we get:

\(\displaystyle \frac{4}{4\sqrt{2}} = \frac{1}{\sqrt{2}}\)

Note: We simplified further because both the numerator and denominator had a "4" which canceled out. 

Now we want to rationalize the denominator,

\(\displaystyle \frac{1}{\sqrt2}\bigg(\frac{\sqrt2}{\sqrt2}\bigg)=\frac{\sqrt2}{2}\)

To simplify, you must use the Law of Exponents.

First you must multiply the coefficients then add the exponents:

\(\displaystyle 2\cdot3\cdot x^{2+5+3}=6\cdot x^{10}=6x^{10}\). 

First, simplify \(\displaystyle \sqrt{12}=\sqrt{4\cdot3}\) to \(\displaystyle 2\sqrt{3}\).

Then set up the multiplication problem:

 \(\displaystyle 2\sqrt{3}\cdot 3\sqrt{6}\).

Multiply the terms outside of the radical, then the terms under the radical:

 \(\displaystyle 2\cdot 3\sqrt{3\cdot6}\) then simplfy: \(\displaystyle 6\sqrt{18}.\) 

The radical is still not in its simplest form and must be reduced further: 

\(\displaystyle 6\sqrt{18}=6\cdot3\sqrt{2}=18\sqrt{2}\). This is the radical in its simplest form. 

To divide the radicals, simply divide the numbers under the radical and leave them under the radical: 

\(\displaystyle \sqrt{\frac{117}{13}}=\sqrt9\) 

Then simplify this radical: 

\(\displaystyle \sqrt9=3\).

When multiplying radicals, just take the values inside the radicand and perfom the operation.

\(\displaystyle \sqrt{2}\cdot \sqrt{3}=\sqrt{2\cdot 3}=\sqrt{6}\)

\(\displaystyle \sqrt{6}\) can't be reduced so this is the final answer.