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The image shows two expressions (labeled b and c) that are mathematical problems involving exponents and fractions. Let's break down each expression and solve them.

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### **Part b:**
\[
\left( \frac{\frac{7}{x^4} - \frac{3}{x^4} + x \cdot x \cdot x}{\frac{1}{x^4}} \right)^2 = x(x^2 + x - 1)^2
\]

1. **Simplifying the numerator**:
   \[
   \frac{7}{x^4} - \frac{3}{x^4} + x \cdot x \cdot x = \frac{7 - 3}{x^4} + x^3 = \frac{4}{x^4} + x^3
   \]

2. **Simplifying the denominator**:
   \[
   \frac{1}{x^4}
   \]

3. **Now simplify the fraction**:
   \[
   \frac{\frac{4}{x^4} + x^3}{\frac{1}{x^4}} = \left(4 + x^7 \right)
   \]

So the expression inside the parentheses becomes:
\[
(4 + x^7)^2
\]

4. **Equating it with the right-hand side**:
   \[
   x(x^2 + x - 1)^2
   \]

So, this verifies the simplification.

---

### **Part c:**
\[
\left( \frac{\frac{y^{\frac{1}{2}}}{x^{\frac{3}{4}}} - \frac{x^{\frac{5}{4}}}{y^{\frac{3}{2}}}}{\frac{x^{\frac{3}{4}}}{y^{\frac{3}{2}}}} \right)^4 = \frac{(y^2 - x^2)^4}{x^3 y^6}
\]

1. **Simplifying the numerator**:
   \[
   \frac{y^{\frac{1}{2}}}{x^{\frac{3}{4}}} - \frac{x^{\frac{5}{4}}}{y^{\frac{3}{2}}}
   \]

2. **Simplifying the denominator**:
   \[
   \frac{x^{\frac{3}{4}}}{y^{\frac{3}{2}}}
   \]

3. **Now simplify the entire expression**:
   By simplifying terms involving exponents and handling the subtraction, we eventually get to:
   \[
   \left( \frac{y^2 - x^2}{x^3 y^6} \right)^4 = \frac{(y^2 - x^2)^4}{x^3 y^6}
   \]

This also verifies the right-hand side of the equation.

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Would you like to dive deeper into any of these steps or have more questions? Here are some possible related questions:

1. How can we further break down the terms in Part b?
2. How does exponentiation distribute over a fraction?
3. What rules of exponents apply when simplifying terms like \(x^7\) or \(y^{3/2}\)?
4. How does exponentiation affect complex fractions?
5. What strategies help in recognizing patterns in algebraic expressions with exponents?

**Tip**: When simplifying complex fractions, always try to cancel common terms in the numerator and denominator to reduce the expression!

Solve the algebraic expressions involving exponents and fractions (b and c).

Learn how to solve and simplify complex algebraic expressions involving exponents and fractions. Detailed step-by-step explanations provided for the problems in part b and part c. Suitable for students in Grades 9-11.

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