https://file.aiquickdraw.com/mathbot/file/1726199460690l8tsvn9t.jpg

Let's solve each problem step-by-step.

### Simplification of Square Roots
1. **\( \sqrt{108} \)**
   - Prime factorization of 108: \(108 = 2^2 \times 3^3\)
   - \( \sqrt{108} = \sqrt{2^2 \times 3^3} = 2 \times 3 \sqrt{3} = 6 \sqrt{3} \)

2. **\( \sqrt{72} \)**
   - Prime factorization of 72: \(72 = 2^3 \times 3^2\)
   - \( \sqrt{72} = \sqrt{2^3 \times 3^2} = 6 \sqrt{2} \)

3. **\( \sqrt{200} \)**
   - Prime factorization of 200: \(200 = 2^3 \times 5^2\)
   - \( \sqrt{200} = \sqrt{2^3 \times 5^2} = 10 \sqrt{2} \)

4. **\( \sqrt{70} \)**
   - Prime factorization of 70: \(70 = 2 \times 5 \times 7\)
   - No perfect square factors, so \( \sqrt{70} \) remains \( \sqrt{70} \)

5. **\( 5 \sqrt{294} \)**
   - Prime factorization of 294: \(294 = 2 \times 3 \times 7^2\)
   - \( \sqrt{294} = \sqrt{2 \times 3 \times 7^2} = 7 \sqrt{6} \)
   - \( 5 \sqrt{294} = 5 \times 7 \sqrt{6} = 35 \sqrt{6} \)

6. **\( 7 \sqrt{50} \)**
   - Prime factorization of 50: \(50 = 2 \times 5^2\)
   - \( \sqrt{50} = \sqrt{2 \times 5^2} = 5 \sqrt{2} \)
   - \( 7 \sqrt{50} = 7 \times 5 \sqrt{2} = 35 \sqrt{2} \)

7. **\( 6 \sqrt{96} \)**
   - Prime factorization of 96: \(96 = 2^5 \times 3\)
   - \( \sqrt{96} = \sqrt{2^5 \times 3} = 4 \sqrt{6} \)
   - \( 6 \sqrt{96} = 6 \times 4 \sqrt{6} = 24 \sqrt{6} \)

8. **\( 7 \sqrt{27} \)**
   - Prime factorization of 27: \(27 = 3^3\)
   - \( \sqrt{27} = \sqrt{3^3} = 3 \sqrt{3} \)
   - \( 7 \sqrt{27} = 7 \times 3 \sqrt{3} = 21 \sqrt{3} \)

### U2T1 Review Problems
9. **Find \( ST \) in problem 9:**
   - Given \( R \) to \( T \) is a line segment, with \( RS = -13 + 2x \), \( ST = x - 7 \), and \( RT = 16 \).
   - The equation is: \( RS + ST = RT \)  
   - \( (-13 + 2x) + (x - 7) = 16 \)  
   - Simplify: \( -20 + 3x = 16 \)  
   - Solve for \( x \): \( 3x = 36 \Rightarrow x = 12 \)  
   - \( ST = x - 7 = 12 - 7 = 5 \)

10. **Find \( ST \) in problem 10:**
    - Given \( S \) to \( U \) is a line segment, with \( ST = x + 17 \), \( TU = x + 19 \), and \( SU = 22 \).
    - The equation is: \( ST + TU = SU \)  
    - \( (x + 17) + (x + 19) = 22 \)  
    - Simplify: \( 2x + 36 = 22 \)  
    - Solve for \( x \): \( 2x = -14 \Rightarrow x = -7 \)  
    - \( ST = x + 17 = -7 + 17 = 10 \)

11. **Problem 11:**
    - \( M \) is the midpoint of \( PQ \), \( PM = 7x + 8 \), and \( MQ = 5x + 20 \). Since \( M \) is the midpoint, \( PM = MQ \).
    - Set the expressions equal: \( 7x + 8 = 5x + 20 \)
    - Solve for \( x \):  
    - \( 7x - 5x = 20 - 8 \Rightarrow 2x = 12 \Rightarrow x = 6 \)  
    - Substitute \( x = 6 \) back:  
    - \( PM = 7(6) + 8 = 42 + 8 = 50 \),  
    - \( MQ = 5(6) + 20 = 30 + 20 = 50 \)

Would you like any further details on these problems, or do you have any questions? 

### Relative Questions:
1. How do you find the prime factorization of a number?
2. What are the steps to simplify a square root?
3. How do you solve equations involving segments on a line?
4. How can the midpoint formula be used in coordinate geometry?
5. Why is it necessary to equate expressions in some algebraic problems?

**Tip:** When simplifying square roots, always look for the largest perfect square factor to reduce the expression efficiently.

Solve the problems related to simplifying square roots and finding the lengths of line segments given algebraic expressions.

This guide offers solutions to simplifying square roots through prime factorization and solving for line segment lengths using algebraic expressions. It is tailored for high school students looking to enhance their mathematical skills.

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