The value of \(\left( {{1 \over 2}} \right)\) [{–2(12 + 2)}10] is:
-140
Evaluating a Mathematical Expression Using Order of Operations
The question asks us to find the value of the given mathematical expression: \(\left( {{1 \over 2}} \right)\) [{–2(12 + 2)}10]. To solve this, we need to follow the order of operations, often remembered by acronyms like PEMDAS or BODMAS.
Understanding Order of Operations
The order of operations dictates the sequence in which mathematical calculations should be performed:
- Parentheses (or Brackets) first
- Exponents (or Orders) next
- Multiplication and Division (from left to right)
- Addition and Subtraction (from left to right)
Step-by-Step Evaluation of the Expression
Let's break down the expression \(\left( {{1 \over 2}} \right)\) [{–2(12 + 2)}10] and evaluate it step by step:
The expression is: \(\left( {{1 \over 2}} \right)\) [{–2(12 + 2)}10]
Step 1: Evaluate the innermost Parentheses/Brackets.
The expression inside the parentheses is (12 + 2).
$$(12 + 2) = 14$$
Now the expression becomes: \(\left( {{1 \over 2}} \right)\) [{–2(14)}10]
Step 2: Perform Multiplication inside the outer Brackets.
Inside the brackets, we have {–2(14)}10. We first perform the multiplication –2 multiplied by 14.
$${–2(14)} = {–2 \times 14} = {–28}$$
Now the expression inside the brackets becomes: [{–28}10]
Step 3: Perform the remaining Multiplication inside the Brackets.
We now have {–28} multiplied by 10.
$${–28}10 = {–28 \times 10} = {–280}$$
The expression now simplifies to: \(\left( {{1 \over 2}} \right)\) [{-280}]
Step 4: Perform the final Multiplication.
Finally, we multiply the result inside the brackets by \(\left( {{1 \over 2}} \right)\).
$$\left( {{1 \over 2}} \right) \times ({-280}) = {1 \over 2} \times {-280}$$
$${1 \over 2} \times {-280} = {-280 \over 2} = {-140}$$
So, the value of the expression \(\left( {{1 \over 2}} \right)\) [{–2(12 + 2)}10] is -140.
Let's summarise the steps in a table:
| Step | Operation | Expression | Result |
|---|---|---|---|
| 1 | Innermost Parentheses | 12 + 2 | 14 |
| 2 | Multiply inside outer Brackets | –2 \(\times\) 14 | –28 |
| 3 | Multiply inside outer Brackets | –28 \(\times\) 10 | –280 |
| 4 | Multiply by factor outside | \(\left( {{1 \over 2}} \right) \times\) –280 | –140 |
The final value obtained is -140.
Revision Table: Key Concepts for Expression Evaluation
| Concept | Description | Importance |
|---|---|---|
| Order of Operations | Specific sequence for calculations (Parentheses/Brackets, Exponents/Orders, Multiplication/Division, Addition/Subtraction). | Ensures a unique and correct result for any mathematical expression. |
| Parentheses/Brackets | Symbols like (), [], {} used to group parts of an expression. Operations inside these are performed first. | Prioritizes certain calculations to be done before others. |
| Fractions | Numbers representing parts of a whole, like \(\left( {{1 \over 2}} \right)\). They act as multipliers or divisors. | Essential for understanding values that are not whole numbers. |
| Negative Numbers | Numbers less than zero. Multiplication rules for positive and negative numbers must be applied correctly. | Crucial for accurate calculations involving signs. |
Additional Information: Applying Order of Operations
The order of operations is a fundamental rule in mathematics that ensures consistency in calculations. Without it, a single expression could have multiple possible values, leading to ambiguity. For example, consider \(2 + 3 \times 4\). If we perform addition first, we get \((2+3) \times 4 = 5 \times 4 = 20\). But if we perform multiplication first, following the rule, we get \(2 + (3 \times 4) = 2 + 12 = 14\). The order of operations tells us that multiplication comes before addition, so 14 is the correct answer.
In our problem, \(\left( {{1 \over 2}} \right)\) [{–2(12 + 2)}10], the brackets [{}] and parentheses () clearly define the groupings. We worked from the inside out:
- Start with the innermost grouping: (12 + 2).
- Move to the next level of grouping: the operations inside the square brackets [{–2(14)}10], following multiplication rules.
- Finally, perform the operation outside the brackets: multiplying by \(\left( {{1 \over 2}} \right)\).
This systematic approach, guided by the order of operations, is key to correctly evaluating such expressions.
Important Questions from Simplification
The value of {5 - 5 ÷ (10 - 12) × 8 + 9} × 3 + 5 + 5 × 5 ÷ 5 of 5 is:
What should come in place of the question mark (?) in the following question?
[((16 ÷ 4) × 4) ÷ 4] = ?
Simplify the following expression.
\(\frac{{6\frac{1}{2} + 2\frac{5}{7} \times \frac{{14}}{{19}} - \frac{1}{2} \div 2\ of\frac{1}{4}}}{{11 \times 12 \div 12 + 12}}\)
The value of \(\left( {\frac{7}{5} \div \frac{7}{{10}}of\frac{3}{4}} \right) \div \frac{4}{9} + \left( {\frac{7}{{16}} \div 10\frac{1}{2} \times 7\frac{1}{5}} \right) \times \frac{5}{{12}} \) is:
- \({{1} \over 3\times4} + {{1} \over 4\times5} + {{1} \over 5\times6} + ... + ...............{{1} \over 20\times21}\) On simplification, the given expression will give the result as:
