Let's work on simplifying each expression given in the image.

Problem 4

The expression is: $\frac{(a^3 b^{-4} c^0)^{-2}}{(a^2 b^5 c^3)^{-4}}$

Step-by-Step Solution

1. Simplify each term in the numerator and the denominator:

   • For $c^0$: Any term raised to the power of zero is 1, so $c^0 = 1$. Thus, $c^0$ can be ignored in the numerator.

   • The expression becomes: $\frac{(a^3 b^{-4})^{-2}}{(a^2 b^5 c^3)^{-4}}$

2. Apply the exponent rule $(x^m)^n = x^{m \cdot n}$:

   • Numerator: $(a^3 b^{-4})^{-2} = a^{3 \cdot -2} b^{-4 \cdot -2} = a^{-6} b^8$
   • Denominator: $(a^2 b^5 c^3)^{-4} = a^{2 \cdot -4} b^{5 \cdot -4} c^{3 \cdot -4} = a^{-8} b^{-20} c^{-12}$

   So the expression now is: $\frac{a^{-6} b^8}{a^{-8} b^{-20} c^{-12}}$

3. Combine the terms by dividing powers with the same base:

   • For $a$: $a^{-6} \div a^{-8} = a^{-6 - (-8)} = a^{2}$
   • For $b$: $b^{8} \div b^{-20} = b^{8 - (-20)} = b^{28}$
   • For $c$: Since $c^{-12}$ is in the denominator, it moves to the numerator as $c^{12}$.

   The simplified expression is: $a^2 b^{28} c^{12}$

Problem 5

The expression is: $\frac{(3x^{-5} y^2)^0}{(4x^{-3} y^2)^{-2}}$

Step-by-Step Solution

1. Simplify each term in the numerator and denominator:

   • The entire numerator $(3x^{-5} y^2)^0$ is raised to the power of zero, so it becomes 1.
   • The expression simplifies to: $\frac{1}{(4x^{-3} y^2)^{-2}}$

2. Apply the exponent rule $(x^m)^n = x^{m \cdot n}$ in the denominator:

   • Denominator: $(4x^{-3} y^2)^{-2} = 4^{-2} x^{(-3) \cdot -2} y^{2 \cdot -2} = \frac{1}{16} x^{6} y^{-4}$

   The expression now is: $1 \cdot \frac{1}{\frac{1}{16} x^{6} y^{-4}} = 16 x^6 y^4$

The simplified form of Problem 5 is: $16 x^6 y^4$

Final Answers

1. For Problem 4: $a^2 b^{28} c^{12}$
2. For Problem 5: $16 x^6 y^4$

Would you like further details on any specific steps, or do you have other questions on these topics?

Here are 5 related questions for further practice:

1. How would you simplify $\frac{(x^2 y^{-3})^3}{(x^{-4} y^2)^2}$?
2. What is the result of $(x^5 y^{-2} z^0)^{-1}$?
3. Simplify $\frac{(a^{-1} b^3)^{-2}}{(a^2 b^{-4})^3}$.
4. How do you simplify expressions with fractional exponents, such as $(x^{1/2} y^{-1/3})^6$?
5. What would the expression $(4x^{-3} y^5)^0$ simplify to, and why?

Tip: When simplifying expressions with exponents, always start by applying any zero exponents and simplifying terms individually before combining them.