https://file.aiquickdraw.com/mathbot/file/1727063744372lqamj766.jpg

Let's go through and solve the equations step by step. I'll break down the problems for clarity.

### Solve each equation:
1. **\( 4 | 1 - 4n | + 6 = 50 \)**
   - Isolate the absolute value term:  
     \( 4 | 1 - 4n | = 50 - 6 \)  
     \( 4 | 1 - 4n | = 44 \)  
     \( | 1 - 4n | = \frac{44}{4} = 11 \)  
   - Set up two cases:  
     1st case: \( 1 - 4n = 11 \)  
     2nd case: \( 1 - 4n = -11 \)  
   - Solve each case:  
     1st case:  
     \( 1 - 4n = 11 \)  
     \( -4n = 10 \)  
     \( n = -\frac{5}{2} \)  
     2nd case:  
     \( 1 - 4n = -11 \)  
     \( -4n = -12 \)  
     \( n = 3 \)

   Solution: \( n = -\frac{5}{2} \) or \( n = 3 \).

2. **\( 7 | -7 - k | - 5 = 16 \)**
   - Isolate the absolute value:  
     \( 7 | -7 - k | = 16 + 5 \)  
     \( 7 | -7 - k | = 21 \)  
     \( | -7 - k | = \frac{21}{7} = 3 \)  
   - Set up two cases:  
     1st case: \( -7 - k = 3 \)  
     2nd case: \( -7 - k = -3 \)  
   - Solve each case:  
     1st case:  
     \( -7 - k = 3 \)  
     \( -k = 10 \)  
     \( k = -10 \)  
     2nd case:  
     \( -7 - k = -3 \)  
     \( -k = 4 \)  
     \( k = -4 \)

   Solution: \( k = -10 \) or \( k = -4 \).

3. **\( 5 | 8k + 10 | - 7 = 43 \)**
   - Isolate the absolute value:  
     \( 5 | 8k + 10 | = 43 + 7 \)  
     \( 5 | 8k + 10 | = 50 \)  
     \( | 8k + 10 | = \frac{50}{5} = 10 \)  
   - Set up two cases:  
     1st case: \( 8k + 10 = 10 \)  
     2nd case: \( 8k + 10 = -10 \)  
   - Solve each case:  
     1st case:  
     \( 8k = 0 \)  
     \( k = 0 \)  
     2nd case:  
     \( 8k = -20 \)  
     \( k = -\frac{5}{2} \)

   Solution: \( k = 0 \) or \( k = -\frac{5}{2} \).

4. **\( 4 | -4x + 5 | + 4 = 56 \)**
   - Isolate the absolute value:  
     \( 4 | -4x + 5 | = 56 - 4 \)  
     \( 4 | -4x + 5 | = 52 \)  
     \( | -4x + 5 | = \frac{52}{4} = 13 \)  
   - Set up two cases:  
     1st case: \( -4x + 5 = 13 \)  
     2nd case: \( -4x + 5 = -13 \)  
   - Solve each case:  
     1st case:  
     \( -4x = 8 \)  
     \( x = -2 \)  
     2nd case:  
     \( -4x = -18 \)  
     \( x = \frac{9}{2} \)

   Solution: \( x = -2 \) or \( x = \frac{9}{2} \).

5. **\( 7 | x - 10 | - 1 = 97 \)**
   - Isolate the absolute value:  
     \( 7 | x - 10 | = 97 + 1 \)  
     \( 7 | x - 10 | = 98 \)  
     \( | x - 10 | = \frac{98}{7} = 14 \)  
   - Set up two cases:  
     1st case: \( x - 10 = 14 \)  
     2nd case: \( x - 10 = -14 \)  
   - Solve each case:  
     1st case:  
     \( x = 24 \)  
     2nd case:  
     \( x = -4 \)

   Solution: \( x = 24 \) or \( x = -4 \).

6. **\( | m - 1 | - 9 = 55 \)**
   - Isolate the absolute value:  
     \( | m - 1 | = 55 + 9 \)  
     \( | m - 1 | = 64 \)  
   - Set up two cases:  
     1st case: \( m - 1 = 64 \)  
     2nd case: \( m - 1 = -64 \)  
   - Solve each case:  
     1st case:  
     \( m = 65 \)  
     2nd case:  
     \( m = -63 \)

   Solution: \( m = 65 \) or \( m = -63 \).

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### Simplify. Your answer should contain only positive exponents:
7. **\( 3x^2 y^3 \cdot 3x^0 y^{-3} \cdot 2x^2 y^3 \)**
   - Combine like terms (use exponent addition):
     \[ 3 \cdot 3 \cdot 2 = 18 \]
     \( x^{2+0+2}

Solve the equations and simplify expressions in a college algebra review worksheet.

This review covers solving absolute value equations and simplifying exponential expressions. Learn step-by-step how to handle absolute value operations and apply exponent rules, essential for College Algebra students.

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