I can see the math problems from the image you uploaded. Let's go through them one by one.

Question 1:

Given $x = -2$, $y = 2$, and $z = 3$, solve:

$\frac{\left( xy^3 \right)^2 \cdot \left( yz^{-1} \right)^{-3}}{xy^2 z^2}$

Question 2:

Simplify the expression:

$\frac{15x^5 yz^{12}}{5x^{-2}z^2}$

Question 3:

Solve the expression:

$1000^0 + 100^0 + 10^0 + 1^0 + \left( \frac{1}{10} \right)^0$

Question 4:

Find the value of:

$81^6 \times 9^5 \div 27^{10}$

Question 5:

Calculate:

$125^{\frac{2}{3}} - 81^{\frac{3}{4}} + 36^{\frac{1}{2}}$

I'll begin by solving these step-by-step.Here are the solutions to the questions:

Question 1:

The result of the expression $\frac{\left( xy^3 \right)^2 \cdot \left( yz^{-1} \right)^{-3}}{xy^2 z^2}$ with $x = -2$, $y = 2$, and $z = 3$ is:

$\boxed{-12}$

Question 2:

The simplified form of $\frac{15x^5 yz^{12}}{5x^{-2} z^2}$ is:

$\boxed{3x^7 yz^{10}}$

Question 3:

The result of the expression $1000^0 + 100^0 + 10^0 + 1^0 + \left(\frac{1}{10}\right)^0$ is:

$\boxed{5}$

Question 4:

The value of $81^6 \times 9^5 \div 27^{10}$ is:

$\boxed{81}$

Question 5:

The result of the expression $125^{\frac{2}{3}} - 81^{\frac{3}{4}} + 36^{\frac{1}{2}}$ is approximately:

$\boxed{4}$

Let me know if you need further details or explanations!

Related Questions:

1. How do negative exponents affect the value of a fraction?
2. What is the property of any number raised to the power of 0?
3. How do you simplify expressions involving powers of products?
4. How do you handle fractional exponents when simplifying roots?
5. What is the best approach to solve mixed exponent expressions?

Tip: Always simplify powers step-by-step by following the order of operations (PEMDAS).