The expression provided in the image is:

$\frac{(16x^{-7}y^3)^{-3} \cdot (4x^7y^{-4})^2}{(2y^{-1}x^4)^{-4}}$

We need to simplify this expression using the properties of exponents and express the answer using positive exponents.

Step 1: Apply the Power of a Product Rule

First, apply the rule $(a \cdot b)^m = a^m \cdot b^m$ to each term:

1. $(16x^{-7}y^3)^{-3}$ becomes $(16)^{-3} \cdot (x^{-7})^{-3} \cdot (y^3)^{-3}$
2. $(4x^7y^{-4})^2$ becomes $4^2 \cdot (x^7)^2 \cdot (y^{-4})^2$
3. $(2y^{-1}x^4)^{-4}$ becomes $(2)^{-4} \cdot (y^{-1})^{-4} \cdot (x^4)^{-4}$

Step 2: Simplify Each Term

Now, simplify each of these terms:

1. $(16)^{-3} = 16^{-3} = (2^4)^{-3} = 2^{-12}$

   • $(x^{-7})^{-3} = x^{21}$ (multiply the exponents: $-7 \times -3 = 21$)
   • $(y^3)^{-3} = y^{-9}$

   So, $(16x^{-7}y^3)^{-3} = 2^{-12} \cdot x^{21} \cdot y^{-9}$

2. $4^2 = 16 = 2^4$

   • $(x^7)^2 = x^{14}$ (multiply the exponents: $7 \times 2 = 14$)
   • $(y^{-4})^2 = y^{-8}$

   So, $(4x^7y^{-4})^2 = 2^4 \cdot x^{14} \cdot y^{-8}$

3. $(2)^{-4} = 2^{-4}$

   • $(y^{-1})^{-4} = y^4$
   • $(x^4)^{-4} = x^{-16}$

   So, $(2y^{-1}x^4)^{-4} = 2^{-4} \cdot y^4 \cdot x^{-16}$

Step 3: Combine the Terms

Now combine the simplified terms into the fraction:

$\frac{2^{-12} \cdot x^{21} \cdot y^{-9} \cdot 2^4 \cdot x^{14} \cdot y^{-8}}{2^{-4} \cdot y^4 \cdot x^{-16}}$

Step 4: Simplify the Expression

Combine like terms:

• Combine the powers of 2:

  • $2^{-12} \cdot 2^4 = 2^{-8}$
  • In the denominator: $2^{-8} \div 2^{-4} = 2^{-4}$

• Combine the powers of $x$:

  • $x^{21} \cdot x^{14} = x^{35}$
  • In the denominator: $x^{35} \cdot x^{-16} = x^{51}$

• Combine the powers of $y$:

  • $y^{-9} \cdot y^{-8} = y^{-17}$
  • In the denominator: $y^{-17} \div y^4 = y^{-21}$

So the expression simplifies to:

$\frac{2^{-8} \cdot x^{35} \cdot y^{-17}}{2^{-4} \cdot x^{-16} \cdot y^4} = 2^{-4} \cdot x^{51} \cdot y^{-21}$

Step 5: Express with Positive Exponents

To express the answer with positive exponents:

$\frac{x^{51}}{16 \cdot y^{21}}$

This is the simplified expression with all positive exponents:

$\boxed{\frac{x^{51}}{16y^{21}}}$

Would you like any more details on a specific step or have any other questions?

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Here are five related questions that can expand on this topic:

1. How do you simplify expressions when dealing with fractional exponents?
2. What are the key steps to simplifying expressions with multiple variables and exponents?
3. How does one deal with negative exponents in complex expressions?
4. Can you demonstrate how to handle expressions involving roots and exponents together?
5. How can the power of a quotient rule be applied to simplify rational expressions?

Tip: When dealing with negative exponents, remember that $a^{-n} = \frac{1}{a^n}$. This will help in transforming the expression into positive exponents.