In the denominator, apply the exponent to the terms in parentheses.

\(\displaystyle \frac{2x^{-7}}{(3x)^{-2}}=\frac{2x^{-7}}{(3^{-2}) (x^{-2})}\)

Terms with negative exponents can be inverted within the fraction to eliminate the negative.

\(\displaystyle \frac{2x^{-7}}{(3^{-2}) (x^{-2})}=\frac{2(3^2)(x^2)}{x^7}\)

Simplify by canceling variables and mulitplying.

\(\displaystyle \frac{2(3^2)(x^2)}{x^7}=\frac{18x^2}{x^7}=\frac{18}{x^5}\)

Break up the fraction into components with like-terms.

\(\displaystyle \frac{8x^{-2a}y^{2a}}{4x^ay^{-2a}}=\frac{8}{4}\times \frac{x^{-2a}}{x^a}\times \frac{y^{2a}}{y^{-2a}}\)

Simplify by combining exponents. Remember that exponents in the denominator will become negative.

\(\displaystyle 2\times x^{(-2a-a)}\times y^{(2a+2a)}\)

\(\displaystyle 2x^{-3a}y^{4a}\)

Finally, put any negative exponent terms into the denominator.

\(\displaystyle \frac{2y^{4a}}{x^{3a}}\)

\(\displaystyle \frac{(2xy)^{6}}{2x^{2} y^{8}}\)

\(\displaystyle = \frac{2^{6} \cdot x^{6}y^{6}}{2x^{2} y^{8}}\)

\(\displaystyle = \frac{64 x^{6-2}}{2 y^{8-6}}\)

\(\displaystyle = \frac{32 x^{4} }{ y^{2}}\)

To divide variables with exponents, we will use the following formula:

\(\displaystyle x^a \div x^b = x^{a-b}\)

Now, let’s combine the following:

\(\displaystyle x^8 \cdot x^2 = x^{8-2}\)

\(\displaystyle x^8 \cdot x^2 = x^{6}\)

When dividing variables, we can only divide like variable.

Also, when dividing variables with exponents, we subtract the exponents.  In other words:

\(\displaystyle \frac{a^x}{a^y} = a^{x-y}\)

So, we get

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

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

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

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

\(\displaystyle ab^2\)

To simplify, we will use the following formula:

\(\displaystyle x^a \div x^b = x^{a-b}\)

So, given the problem, we get

\(\displaystyle y^{12} \div y^2 = y^{12-2}\)

\(\displaystyle y^{12} \div y^2 = y^{10}\)

To divide like terms with exponents, we will use the following formula:

\(\displaystyle \frac{a^x}{a^y} = a^{x-y}\)

So, we get

\(\displaystyle \frac{h^{10}}{h^2}\)

\(\displaystyle h^{10-2}\)

\(\displaystyle h^8\)

To divide like variables with exponents, we will use the following formula:

\(\displaystyle \frac{a^x}{a^y} = a^{x-y}\)

So, we get

\(\displaystyle \frac{y^9}{y^3} = y^{9-3}\)

\(\displaystyle \frac{y^9}{y^3} = y^6\)

To divide variables with exponents, we will use the following formula:

\(\displaystyle x^a \div x^b = x^{a-b}\)

Now, let’s divide the following:

\(\displaystyle p^9 \div p^3 = p^{9-3}\)

\(\displaystyle p^9 \div p^3 = p^{6}\)

To divide like variables with exponents, we will use the following formula:

\(\displaystyle \frac{a^x}{a^y} = a^{x-y}\)

Also, we divide the coefficients like normal. 

So, we get

\(\displaystyle \frac{18x^4}{9x}\)

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

\(\displaystyle \frac{2x^4}{x}\)

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

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

\(\displaystyle 2x^3\)