Simplifying Mathematical Expressions Using Order of Operations

Simplifying mathematical expressions requires following a specific order of operations to ensure a unique and correct result. This order is often remembered using acronyms like BODMAS or PEMDAS.

Understanding the Order of Operations (BODMAS/PEMDAS)

The order of operations dictates the sequence in which different operations should be performed in an expression:

• Brackets (Parentheses, braces, vinculum)
• Orders (Exponents, roots)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

In the given expression, we have brackets (braces and the vinculum or overline) and arithmetic operations (subtraction, multiplication, addition).

The given expression is:

\(71-(-3)\{-2-(\overline{8-3}\}+3\{5+(-2)(-1)\}\)

Step-by-Step Simplification of the Expression

Step 1: Address the Vinculum (Overline)

The vinculum acts like a grouping symbol. We first evaluate the operation under the vinculum:

\(\overline{8-3} = 5\)

Substitute this value back into the expression:

\(71-(-3)\{-2-(5\}+3\{5+(-2)(-1)\}\)

Step 2: Evaluate Expressions Inside Parentheses/Braces

Next, we evaluate the expressions within the innermost grouping symbols (parentheses inside braces) and then the expressions within the braces.

First set of braces: \(\{-2-(5\}\)

Evaluate inside this brace:

\(-2 - 5 = -7\)

Second set of braces: \(\{5+(-2)(-1)\}\)

Inside this brace, we have multiplication and addition. Perform the multiplication first:

\((-2)(-1) = 2\)

Now perform the addition inside the second brace:

\(5 + 2 = 7\)

Substitute these values back into the expression:

\(71-(-3)\{-7\}+3\{7\}\)

Step 3: Perform Multiplication

Now we have multiplication operations outside the remaining braces. We have \(-(-3)\{-7\}\) and \(+3\{7\}\).

First multiplication: \(-(-3)\{-7\}\)

This can be written as \(-1 \times (-3) \times (-7)\). Multiplying \(-1\) and \(-3\) gives \(3\). Then multiplying \(3\) by \(-7\) gives \(-21\).

\(-(-3)\{-7\} = (3)(-7) = -21\)

Second multiplication: \(+3\{7\}\)

This is \(3 \times 7 = 21\).

Substitute these results back into the expression:

\(71 - 21 + 21\)

Step 4: Perform Addition and Subtraction

Finally, perform addition and subtraction from left to right.

\(71 - 21 = 50\)

Now add 21:

\(50 + 21 = 71\)

The simplified value of the expression is 71.

Let's summarize the steps in a table:

Following the order of operations correctly, the expression simplifies to 71.

Revision Table: Order of Operations Key Concepts

Additional Information: Handling Signs in Expressions

When simplifying expressions, paying close attention to positive and negative signs is crucial, especially during multiplication and subtraction.

• Subtracting a negative number: Subtracting a negative is the same as adding the positive equivalent. Example: \(a - (-b) = a + b\). In our problem, \(71 - (-3)\) became \(71 + 3\) effectively, although the multiplication \(-(-3)\{-7\}\) was handled together. The \(-(-3)\) part before multiplying by \(\{-7\}\) is \(-1 \times (-3) = 3\).
• Multiplying signs:
  • Positive \(\times\) Positive = Positive
  • Negative \(\times\) Negative = Positive (like \((-2)(-1)=2\))
  • Positive \(\times\) Negative = Negative (like \((3)(-7)=-21\))
  • Negative \(\times\) Positive = Negative

Careful application of these rules is vital for accurate simplification.