https://file.aiquickdraw.com/mathbot/file/17255994173566ls56pqv.jpg

I have opened the image. I will now proceed to solve questions 4 to 6.

### Problem 4:
The problem asks to simplify the following expressions:

- \( a. \ 5 \times 10^9 \)
- \( b. \ 0.05 \times 10^7 \times (2 \times 10^3) \)
- \( c. \ (2.5 \times 10^7) \div (0.5 \times 10^2) \)
- \( d. \ 24 \times 10^7 \div 1,000,000 \)

#### Solutions:
1. \( 5 \times 10^9 \) is already in simplified form.
   
2. \( 0.05 \times 10^7 \times (2 \times 10^3) = (0.05 \times 2) \times 10^{7+3} = 0.1 \times 10^{10} = 1 \times 10^9 \).
   
3. \( (2.5 \times 10^7) \div (0.5 \times 10^2) = (2.5 \div 0.5) \times 10^{7-2} = 5 \times 10^5 \).
   
4. \( 24 \times 10^7 \div 1,000,000 = 24 \times 10^7 \div 10^6 = 24 \times 10^{7-6} = 24 \times 10^1 = 240 \).

---

### Problem 5:
Here, given \( R = 25 \ \text{cm} \) and \( I = 243 \), find the expression for:
- \( a. \  \frac{R^2}{I^2} \)
- \( b. \  \frac{R^{2/3}}{I^{4/3}} \)
- \( c. \  \left( \frac{R}{I} \right)^{3/2} \)

#### Solutions:
1. \( \frac{R^2}{I^2} = \left( \frac{25}{243} \right)^2 = \frac{625}{59049} \).

2. \( \frac{R^{2/3}}{I^{4/3}} = \left( \frac{25}{243} \right)^{2/3} = \frac{25^{2/3}}{243^{4/3}} \). This requires calculating the cube roots:
   - \( 25^{2/3} = 5^{4/3} \), and \( 243^{4/3} = 3^8 = 6561 \), so:
   - \( \frac{25^{2/3}}{243^{4/3}} = \frac{25^{2/3}}{6561} \).

3. \( \left( \frac{R}{I} \right)^{3/2} = \left( \frac{25}{243} \right)^{3/2} = \frac{25^{3/2}}{243^{3/2}} \). This gives:
   - \( 25^{3/2} = 5^3 = 125 \), and \( 243^{3/2} = 3^9 = 19683 \), so:
   - \( \frac{25^{3/2}}{243^{3/2}} = \frac{125}{19683} \).

---

### Problem 6:
This problem asks to simplify:
- \( \sqrt{\frac{8}{5}} \div \sqrt{\frac{7}{5}} \)
- Options:
  a. \( \frac{\sqrt{8}}{\sqrt{5}} \)
  b. \( \frac{5}{\sqrt{8}} \)
  c. \( \frac{\sqrt{8}}{\sqrt{7}} \)
  d. \( \frac{\sqrt{7}}{\sqrt{5}} \)
  e. \( \frac{\sqrt{8}}{5} \)

#### Solution:
\[
\sqrt{\frac{8}{5}} \div \sqrt{\frac{7}{5}} = \frac{\sqrt{8/5}}{\sqrt{7/5}} = \sqrt{\frac{8}{7}}
\]
Thus, the correct simplification is \( \frac{\sqrt{8}}{\sqrt{7}} \), which matches option **c**.

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Let me know if you would like further details or have any questions.

**Relative Questions:**
1. What are other methods to simplify scientific notation like in problem 4?
2. How do you calculate cube roots in expressions like problem 5b?
3. What are some rules of exponents that are useful for expressions in problems like 5c?
4. How do you apply rationalization when dividing square roots as seen in problem 6?
5. What are some common simplifications for radicals involving fractions?

**Tip**: When simplifying radical expressions, always try to rationalize the denominator to express the result in the simplest form.

Explore how to solve advanced mathematical problems involving exponentiation, scientific notation, and radicals with detailed examples. Learn the techniques for simplifying complex expressions and applying exponent laws effectively.

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