Each variable in this expression can be solved separately.  When we are dividing the powers of the same base, we can simply subtract the powers.

\(\displaystyle \frac{x^3}{x^5} = x^{3-5} = x^{-2}\)

The negative exponent can be converted to:  \(\displaystyle \frac{1}{x^2}\)

Dividing the \(\displaystyle y\) terms on the numerator and denominator will give us \(\displaystyle 1\).

\(\displaystyle \frac{y}{y} = \frac{y^1}{y^1} = y^{1-1}= y^0 = 1\)

Divide \(\displaystyle \frac{z^2}{z^3}\) and simplify to a fraction.

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

Multiply all the terms.

\(\displaystyle (\frac{1}{x^2})(1)(\frac{1}{z}) = \frac{1}{x^2z}\)

The answer is:  \(\displaystyle \frac{1}{x^2z}\)

Write out the factors for the numerator and denominator.

\(\displaystyle \frac{3xy^2}{9x^2y^4}= \frac{3 \cdot x \cdot y^2}{3\cdot3\cdot x \cdot x \cdot y^2 \cdot y^2}\)

Cancel the common factors on the top and bottom.

The answer is:  \(\displaystyle \frac{1}{3xy^2}\)

Given the problem 

\(\displaystyle \frac{8a^2b^3}{-2ab}\)

we will write it like

\(\displaystyle (\frac{8}{-2})(\frac{a^2b^3}{ab})\)

We can easily simplify the first part

\(\displaystyle (-4)(\frac{a^2b^3}{ab})\)

Now, when dividing, to simplify monomials containing the same base, you substract their exponents.  We get

\(\displaystyle (-4)(a^{2-1}b^{3-1})\)

\(\displaystyle (-4)(a^1b^2)\)

\(\displaystyle (-4)(ab^2)\)

\(\displaystyle -4ab^2\)

In order to solve this expression, we need to write out the factors of the numerator and denominator.

\(\displaystyle \frac{5x^2y}{15xy^6} = \frac{5\cdot x \cdot x \cdot y}{5 \cdot 3 \cdot x \cdot y \cdot y^5}\)

We can then cancel out the common terms.

The answer is:  \(\displaystyle \frac{x}{3y^5}\)

In order to divide, we should split the top and bottom into their factors.

\(\displaystyle \frac{6ab^2}{6b^3} = \frac{6 \times a \times b^2}{6 \times b \times b^2}\)

The six and \(\displaystyle b^2\) terms can be cancelled in the numerator and denominator.

The answer is:  \(\displaystyle \frac{a}{b}\)

When similar bases of a certain power are divided, their powers can be subtracted.

Evaluate each term.

\(\displaystyle \frac{18}{36} = \frac{1}{2}\)

When dealing with negative exponents, they can be rewritten as the reciprocal of the positive exponent.

\(\displaystyle \frac{x^2}{x^5} = x^{2-5} = x^{-3} = \frac{1}{x^3}\)

\(\displaystyle \frac{y^3}{y^{-3}} = y^{3-(-3)} = y^6\)

Multiply all the terms together.

\(\displaystyle \frac{1}{2}\cdot\frac{1}{x^3}\cdot y^6 = \frac{y^6}{2x^3}\)

The answer is:  \(\displaystyle \frac{y^6}{2x^3}\)

Rewrite both the numerator and denominator by factors.

\(\displaystyle \frac{3x^3y}{6xy^2z} = \frac{3 \times x \times x \times x \times y}{3 \times 2 \times y \times y \times z}\)

Cancel all the common terms on the top and bottom.

The answer is:  \(\displaystyle \frac{x^2}{2yz}\)

Some of the terms of the given expression can already be simplified.

Cancel out the \(\displaystyle 3x^2\) terms.

\(\displaystyle \frac{y^5z}{y^{-5}z^3}\)

When we are dividing powers of a similar base, it is the same as subtracting the exponents.

\(\displaystyle \frac{y^5}{y^{-5}} = y^{5-(-5)}= y^{10}\)

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

A negative exponent can be written as one over the base's positive exponent.  

Multiply both terms together.

\(\displaystyle y^{10} \cdot \frac{1}{z^2}= \frac{y^{10}}{z^2}\)

The answer is:  \(\displaystyle \frac{y^{10}}{z^2}\)

In order to simplify this fraction,  we will need to rewrite the numerator using factors.

\(\displaystyle \frac{8xz^3}{2z}=\frac{2\cdot 4 \cdot x \cdot z^3}{2z}\)

Cancel the common terms in the numerator and denominator.

The answer is:  \(\displaystyle 4xz^2\)

When powers of a similar base are divided, the powers can be subtracted.

The coefficients will cancel.

Subtract each power for each variable.

\(\displaystyle x^{6-(-6)}=x^{12}\)

\(\displaystyle y^{1-6}= y^{-5} = \frac{1}{y^5}\)

\(\displaystyle z^{3-5}= z^{-2} = \frac{1}{z^2}\)

Multiply and combine each of these terms together.

\(\displaystyle x^{12} \times \frac{1}{y^5} \times \frac{1}{z^2}\)

The answer is:  \(\displaystyle \frac{x^{12}}{y^5z^2}\)