https://file.aiquickdraw.com/mathbot/file/1726687603117brjpzjgm.jpg

The image contains two mathematical expressions labeled (ii) and (iii). Let's go over both expressions step-by-step:

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### (ii) Expression:

\[
\left( \frac{x^{\frac{7}{4}} - x^{\frac{3}{4}} + x \cdot x^{\frac{7}{4}}}{x^4} \right)^2
\]

- **Numerator:**
  \[
  x^{\frac{7}{4}} - x^{\frac{3}{4}} + x \cdot x^{\frac{7}{4}} = x^{\frac{7}{4}} - x^{\frac{3}{4}} + x^{1 + \frac{7}{4}} = x^{\frac{7}{4}} - x^{\frac{3}{4}} + x^{\frac{11}{4}}
  \]

- **Denominator:**
  \( x^4 \)

Thus, the expression simplifies to:

\[
\left( \frac{x^{\frac{7}{4}} - x^{\frac{3}{4}} + x^{\frac{11}{4}}}{x^4} \right)^2
\]

Now, distribute the denominator \( x^4 \) to each term in the numerator:

\[
= \left( x^{\frac{7}{4} - 4} - x^{\frac{3}{4} - 4} + x^{\frac{11}{4} - 4} \right)^2
= \left( x^{-\frac{9}{4}} - x^{-\frac{13}{4}} + x^{-\frac{5}{4}} \right)^2
\]

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### (iii) Expression:

\[
\left[ \frac{y^{\frac{1}{2}} - x^{\frac{5}{4}}}{x^{\frac{3}{4}} - y^{\frac{3}{2}}} \right]^4
\]

- **Numerator:**
  \( y^{\frac{1}{2}} - x^{\frac{5}{4}} \)

- **Denominator:**
  \( x^{\frac{3}{4}} - y^{\frac{3}{2}} \)

The expression stays as is but can be simplified if needed by combining like terms. However, it doesn't simplify further at this stage without factoring or more context.

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Let me know if you'd like further simplifications or explanations!

### Relative Questions:
1. How do you simplify exponents when they involve fractions?
2. What rules govern the multiplication of terms with the same base?
3. Can the third expression be factored further?
4. How do negative exponents change the form of an expression?
5. How can we apply the power rule to handle fractional exponents?

### Tip:
When dealing with fractional exponents, it’s useful to remember that \( a^{\frac{m}{n}} = \sqrt[n]{a^m} \), which can make simplification or solving equations easier.

Simplify and analyze the following mathematical expressions: (ii) \left( \frac{x^{7/4} - x^{3/4} + x \cdot x^{7/4}}{x^4} \right)^2 and (iii) \left[ \frac{y^{1/2} - x^{5/4}}{x^{3/4} - y^{3/2}} \right]^4.

Explore the step-by-step simplification of complex expressions involving fractional exponents and rational expressions, suitable for high school students in grades 9 to 12. This detailed guide demonstrates how to apply exponent rules and simplify complex rational expressions effectively.

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