Simplifying the Mathematical Expression using Order of Operations

We are asked to simplify the given mathematical expression:

\(\left( { - 8} \right)\frac{{\left( {36\; \div \;\left\{ {7\; + \;2} \right\}} \right)}}{{\left( { - 4} \right)\left\{ {19\; - \left( { - 3} \right)\left( { - 5} \right)} \right\}}}\)

To simplify this expression, we must follow the order of operations, commonly known as BODMAS or PEMDAS.

• Brackets (or Parentheses) - Solve expressions inside brackets first.
• Orders (or Exponents/Indices) - Solve powers and roots.
• Division and Multiplication - Solve from left to right.
• Addition and Subtraction - Solve from left to right.

Step-by-Step Simplification of the Numerator

The numerator is \(\left( { - 8} \right)\left( {36\; \div \;\left\{ {7\; + \;2} \right\}} \right)\).

1. First, simplify the expression inside the curly braces: \(\{7 + 2\}\).
   \(\{7 + 2\} = 9\)
2. Now, perform the division inside the parentheses: \(\left( {36\; \div \;9} \right)\).
   \(\left( {36\; \div \;9} \right) = 4\)
3. Finally, perform the multiplication: \(\left( { - 8} \right) \times 4\).
   \(\left( { - 8} \right) \times 4 = -32\)

So, the simplified numerator is \(-32\).

Step-by-Step Simplification of the Denominator

The denominator is \(\left( { - 4} \right)\left\{ {19\; - \left( { - 3} \right)\left( { - 5} \right)} \right\}\).

1. First, simplify the expression inside the curly braces \(\left\{ {19\; - \left( { - 3} \right)\left( { - 5} \right)} \right\}\). Inside these braces, we have multiplication and subtraction. We do multiplication first: \(\left( { - 3} \right)\left( { - 5} \right)\).
   \(\left( { - 3} \right) \times \left( { - 5} \right) = 15\) (Remember: negative multiplied by negative is positive).
2. Now, perform the subtraction inside the curly braces: \(\{19 - 15\}\).
   \(\{19 - 15\} = 4\)
3. Finally, perform the multiplication: \(\left( { - 4} \right) \times 4\).
   \(\left( { - 4} \right) \times 4 = -16\)

So, the simplified denominator is \(-16\).

Final Calculation: Dividing Numerator by Denominator

Now we divide the simplified numerator by the simplified denominator:

\(\frac{{\text{Numerator}}}{{\text{Denominator}}} = \frac{{ - 32}}{{ - 16}}\)

When dividing two numbers with the same sign (both negative in this case), the result is positive.

\(\frac{{ - 32}}{{ - 16}} = \frac{{32}}{{16}} = 2\)

Thus, the simplified value of the expression is 2.

Revision Table: Order of Operations (BODMAS/PEMDAS)

Additional Information on Integer Operations

Understanding how to work with positive and negative integers is crucial for simplifying expressions like this one.

• Addition:
  • Positive + Positive = Positive (e.g., \(3 + 5 = 8\))
  • Negative + Negative = Negative (e.g., \(-3 + (-5) = -8\))
  • Positive + Negative: Subtract the smaller absolute value from the larger absolute value and take the sign of the number with the larger absolute value (e.g., \(5 + (-3) = 2\), \(-5 + 3 = -2\)).
• Subtraction:
  • Subtracting a positive number is the same as adding a negative number (e.g., \(5 - 3 = 5 + (-3) = 2\)).
  • Subtracting a negative number is the same as adding a positive number (e.g., \(5 - (-3) = 5 + 3 = 8\)).
• Multiplication/Division:
  • Positive \(\times\) Positive = Positive
  • Negative \(\times\) Negative = Positive
  • Positive \(\times\) Negative = Negative
  • Negative \(\times\) Positive = Negative
  • The same rules apply for division.