To solve this, remember that when multiplying variables, exponents are added.  When raising a power to a power, exponents are multiplied.  Thus:

\(\displaystyle (x^{2})^{\frac{1}{3}}(x^{-2})^{\frac{1}{2}}(x^{4})^{\frac{1}{3}} =\)

\(\displaystyle x^{\frac{2}{3}}x^{-1}x^{\frac{4}{3}} = x^{\frac{2}{3}-1+\frac{4}{3}} = x\)

Since \(\displaystyle 18 = 2 \cdot 3 ^{2}\), we can multiply 18 by \(\displaystyle 2^{2} \cdot 3 = 12\) to yield the lowest possible perfect cube:

\(\displaystyle 18 \cdot 12 = 216 = 6 ^{3}\)

Therefore, to rationalize the denominator, we multiply both nuerator and denominator by \(\displaystyle \sqrt[3]{12}\) as follows:

\(\displaystyle \frac{6}{\sqrt[3]{18} }\)

\(\displaystyle = \frac{6 \cdot \sqrt[3]{12} }{\sqrt[3]{18} \cdot \sqrt[3]{12} }\)

\(\displaystyle = \frac{6 \cdot \sqrt[3]{12} }{\sqrt[3]{18 \cdot 12 } }\)

\(\displaystyle = \frac{6 \sqrt[3]{12} }{\sqrt[3]{216 } }\)

\(\displaystyle = \frac{6 \sqrt[3]{12} }{6 }\)

\(\displaystyle = \sqrt[3]{12}\)

Begin by getting a prime factor form of the contents of your root.

\(\displaystyle 120000 = 12 * 10000 = 2^2*3*10^4\)

Applying some exponent rules makes this even faster:

\(\displaystyle 10^4 = (2*5)^4=2^45^4\)

Put this back into your problem:

\(\displaystyle 2^2*3*2^4*5^4=2^6*3*5^4\)

Returning to your radical, this gives us:

\(\displaystyle \sqrt[3]{2^6*3*5^4}\)

Now, we can factor out \(\displaystyle 2\) sets of \(\displaystyle 2^3\) and \(\displaystyle 1\) set of \(\displaystyle 5^3\).  This gives us:

\(\displaystyle 2^2*5\sqrt[3]{3*5}=20\sqrt[3]{15}\)

Begin by factoring the contents of the radical:

\(\displaystyle 15625 = 25*625 = 25*25*25=5*5*5*5*5*5=5^6\)

This gives you:

\(\displaystyle \sqrt[4]{5^6}\)

You can take out \(\displaystyle 1\) group of \(\displaystyle 5^4\).  That gives you:

\(\displaystyle 5\sqrt[4]{5^2}=5\sqrt[4]{25}\)

Using fractional exponents, we can rewrite this:

\(\displaystyle 5*5^\frac{2}{4}\)

Thus, we can reduce it to:

\(\displaystyle 5*5^\frac{1}{2}\)

Or:

\(\displaystyle 5\sqrt{5}\)

To simplify \(\displaystyle \sqrt{125}+\sqrt{50}\), find the common factors of both radicals.

\(\displaystyle \sqrt{125}= \sqrt{25\cdot5}=\sqrt{25}\cdot \sqrt5=5\sqrt5\)

\(\displaystyle \sqrt{50}=\sqrt{25\cdot 2}=\sqrt{25}\cdot \sqrt2=5\sqrt2\)

Sum the two radicals.

The answer is:  \(\displaystyle 5\sqrt5+5\sqrt2\)

To take the cube root of the term on the inside of the radical, it is best to start by factoring the inside:

\(\displaystyle \sqrt[3]{x^3\cdot x^2\cdot y^6\cdot 27}\)

Now, we can identify three terms on the inside that are cubes:

\(\displaystyle x^3, y^6, 27\)

We simply take the cube root of these terms and bring them outside of the radical, leaving what cannot be cubed on the inside of the radical.

\(\displaystyle x\cdot y^2 \cdot 3 \cdot \sqrt[3]{x^2}\)

Rewritten, this becomes

\(\displaystyle 3xy^2\sqrt[3]{x^2}\)

Simplify both radicals by rewriting each of them using common factors.

\(\displaystyle \sqrt{32} = \sqrt4 \cdot \sqrt{4}\cdot \sqrt{2} = 4\sqrt2\)

\(\displaystyle \sqrt{8} = \sqrt4 \cdot \sqrt2 = 2\sqrt2\)

Multiply the two radicals.

\(\displaystyle 4\sqrt2\cdot 2\sqrt2 = 4\cdot 2 \cdot 2 = 16\)

The answer is:  \(\displaystyle 16\)

In order to simplify this radical, rewrite the radical using common factors.

\(\displaystyle -\sqrt{ 120} = -\sqrt{2} \cdot \sqrt{2}\cdot \sqrt{5} \cdot \sqrt{6}\)

Simplify the square roots.  

\(\displaystyle -2\cdot \sqrt{5} \cdot \sqrt{6}\)

Multiply the terms inside the radical.

The answer is:  \(\displaystyle -2\sqrt{30}\)

Break down the two radicals by their factors.

\(\displaystyle \sqrt{30} \cdot \sqrt{15} = (\sqrt{3}\times\sqrt{2}\times \sqrt{5})\cdot(\sqrt{3}\times \sqrt{5})\)

A square root of a number that is multiplied by itself is equal to the number inside the radical.

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

\(\displaystyle \sqrt{5}\cdot \sqrt{5}=5\)

Simplify the terms in the parentheses.

\(\displaystyle (\sqrt{3}\times\sqrt{2}\times \sqrt{5})\cdot(\sqrt{3}\times \sqrt{5}) = 3\times 5 \times \sqrt2\)

The answer is:  \(\displaystyle 15\sqrt2\)

The first term is already simplified.  The second and third term will need to be simplified.

Write the common factors of the second radical and simplify.

\(\displaystyle 6\sqrt{50} =6\sqrt{25\times 2} = 6\sqrt{25}\times \sqrt{2} = 6(5)\sqrt2 = 30\sqrt2\)

Repeat the process for the third term.

\(\displaystyle 9\sqrt{500} = 9\sqrt{100\times 5} = 9\sqrt{100}\times \sqrt{5} = 9(10)\sqrt{5} = 90\sqrt5\)

Rewrite the expression.

\(\displaystyle 3\sqrt{5}+6\sqrt{50}+ 9\sqrt{500}= 3\sqrt{5}+30\sqrt2+90\sqrt5\)

Combine like-terms.

The answer is:  \(\displaystyle 30\sqrt2+93\sqrt5\)