Sure, let's simplify each expression step by step.

1. Simplify:
   \[25 - ( \sqrt { 16} - 1) ( 3 - 9) ^ 2\]
   Solution:
   \[\sqrt { 16} = 4\]
   \[3 - 9 = - 6\]
   \[( - 6) ^ 2 = 36\]
   \[25 - ( 4 - 1) ( 36) = 25 - 3 \times 36 = 25 - 108 = - 83\]
   Answer: \(- 83\)
2. Simplify:
   \[\left | 5 - 2( 9) \right | + \frac { 9^ 2} { 3} \]
   Solution:
   \[2( 9) = 18\]
   \[5 - 18 = - 13\]
   \[\left | - 13 \right | = 13\]
   \[9^ 2 = 81\]
   \[\frac { 81} { 3} = 27\]
   \[13 + 27 = 40\]
   Answer: \(40\)
3. Simplify:
   \[\frac { 2^ 5 - 4 \times 3^ 3} { 7 + ( 1 - \sqrt { 100} ) } \]
   Solution:
   \[2^ 5 = 32\]
   \[3^ 3 = 27\]
   \[4 \times 27 = 108\]
   \[32 - 108 = - 76\]
   \[\sqrt { 100} = 10\]
   \[1 - 10 = - 9\]
   \[7 + ( - 9) = - 2\]
   \[\frac { - 76} { - 2} = 38\]
   Answer: \(38\)
4. Simplify:
   \[\frac { ( \sqrt { 225} - 11) \times 12} { - 12 - ( - 8 - 6^ 2) } - \left | - 3 \right | \]
   Solution:
   \[\sqrt { 225} = 15\]
   \[15 - 11 = 4\]
   \[4 \times 12 = 48\]
   \[6^ 2 = 36\]
   \[- 8 - 36 = - 44\]
   \[- 12 - ( - 44) = - 12 + 44 = 32\]
   \[\frac { 48} { 32} = \frac { 3} { 2} \]
   \[\left | - 3 \right | = 3\]
   \[\frac { 3} { 2} - 3 = \frac { 3} { 2} - \frac { 6} { 2} = - \frac { 3} { 2} \]
   Answer: \(- \frac { 3} { 2} \)
5. Simplify:
   \[\frac { 3} { 2} \left [ \frac { 58 - 10^ 2} { \sqrt { 16} \div 3} \right ] \]
   Solution:
   \[10^ 2 = 100\]
   \[58 - 100 = - 42\]
   \[\sqrt { 16} = 4\]
   \[4 \div 3 = \frac { 4} { 3} \]
   \[\frac { - 42} { \frac { 4} { 3} } = - 42 \times \frac { 3} { 4} = - \frac { 126} { 4} = - 31.5\]
   \[\frac { 3} { 2} \times - 31.5 = - 47.25\]
   Answer: \(- 47.25\)
6. Simplify:
   \[\frac { 56 \div ( 7 - 9) ^ 3 - 25} { 23 - 5 \times 4} \]
   Solution:
   \[7 - 9 = - 2\]
   \[( - 2) ^ 3 = - 8\]
   \[56 \div - 8 = - 7\]
   \[- 7 - 25 = - 32\]
   \[5 \times 4 = 20\]
   \[23 - 20 = 3\]
   \[\frac { - 32} { 3} = - \frac { 32} { 3} \]
   Answer: \(- \frac { 32} { 3} \)
7. Evaluate given \(a = - 7\) and \(b = 2\):
   \[a^ 2 + 2( b - 6) - 17\]
   Solution:
   \[( - 7) ^ 2 = 49\]
   \[2( 2 - 6) = 2( - 4) = - 8\]
   \[49 - 8 - 17 = 24\]
   Answer: \(24\)
8. Evaluate given \(x = 4\) and \(y = - 7\):
   \[\frac { 8x - 2y} { 10xy} \]
   Solution:
   \[8( 4) = 32\]
   \[- 2( - 7) = 14\]
   \[32 + 14 = 46\]
   \[10( 4) ( - 7) = - 280\]
   \[\frac { 46} { - 280} = - \frac { 23} { 140} \]
   Answer: \(- \frac { 23} { 140} \)

Supplemental Knowledge

Simplifying algebraic expressions involves combining like terms, using the order of operations (PEMDAS/BODMAS), and applying arithmetic rules. Evaluating expressions requires substituting given values for variables and then simplifying.

1. Order of Operations (PEMDAS/BODMAS):
   • Parentheses/Brackets
   • Exponents/Orders
   • Multiplication and Division (from left to right)
   • Addition and Subtraction (from left to right)
2. Absolute Value:
   • The absolute value of a number is its distance from zero on the number line, always non-negative.
3. Substitution:
   • Replacing variables with given numerical values to evaluate an expression.

Knowledge in Action

Learning to simplify and evaluate expressions is fundamental for numerous real-life applications, from budget calculations and distance measurements in physics, to altering recipes in cooking.

For more help with algebraic simplifications or evaluations check out UpStudy’s Algebra calculator! This tool can assist you in mastering these concepts by providing step-by-step solutions tailored to your learning needs.
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