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.

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

In this case, we have a perfect square so simplify that first.

Then, take that answer and multiply that with \(\displaystyle \sqrt{3}\) to get the final answer.

\(\displaystyle {\sqrt{9}}\cdot \sqrt{3}=3\cdot \sqrt{3}=3\sqrt{3}\).

When dividing radicals, check the denominator to make sure it can be simplified or that there is a radical present that needs to be fixed. Since there is a radical present, we need to eliminate that radical. To do this, we multiply both top and bottom by \(\displaystyle \sqrt{3}\). The reason is because we want a whole number in the denominator and multiplying by itself will achieve that. By multiplying itself, it creates a square number which can be reduced to \(\displaystyle 3\).

\(\displaystyle \frac{\sqrt{5}}{\sqrt{3}}\cdot \frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt{15}}{\sqrt{9}}=\frac{\sqrt{15}}{3}\)

With the denominator being \(\displaystyle 3\), the numerator is \(\displaystyle \sqrt{5\cdot 3}=\sqrt{15}\). Final answer is \(\displaystyle \frac{\sqrt{15}}{3}\).

When dividing radicals, check the denominator to make sure it can be simplified or that there is a radical present that needs to be fixed.

Both \(\displaystyle 9\) and \(\displaystyle 4\) are perfect squares so they can be simplify.

Final answer is

 \(\displaystyle \frac{\sqrt9}{\sqrt4}=\frac{\sqrt{3\cdot 3}}{\sqrt{2\cdot 2}}=\frac{3}{2}\).

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

Since there is a number outside of the radicand, multiply the outside numbers and then the radicand.

\(\displaystyle 3{\sqrt{2}}\cdot 4\sqrt{3}=3\cdot 4\cdot \sqrt{2\cdot 3}=12\sqrt{6}\)

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

Since there is a number outside of the radicand, multiply the outside numbers and then the radicand. 

\(\displaystyle 3\sqrt{6}\cdot 6\sqrt{10}=3\cdot 6\cdot\sqrt{6\cdot10}=18\sqrt{60}\)

Before we say that's the final answer, check the radicand to see that there are no square numbers that can be factored. A \(\displaystyle 4\) can be factored and thats a perfect square. When I divide \(\displaystyle 60\) with \(\displaystyle 4\), I get \(\displaystyle 15\) which doesn't have perfect square factors.

Therefore, our answer becomes

\(\displaystyle 18{\sqrt{60}}=18\sqrt{4\cdot 15}=18\cdot2\sqrt{15}=36\sqrt{15}\). 

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

Since there is a number outside of the radicand, multiply the outside numbers and then the radicand. 

\(\displaystyle 2\sqrt{15}\cdot \sqrt{15}=2\cdot\sqrt{15\cdot15}=2\cdot15=30\)