The correct sequence for evaluating a mathematical expression follows the order of operations, often remembered by the mnemonic PE(MD)(AS). This provides a clear and consistent approach to solving complex expressions.
Order of Operations Explained
The acronym PE(MD)(AS) breaks down the calculation sequence into a priority order:
1. Parentheses
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Calculations within parentheses are always performed first. This allows for grouping and prioritizing specific parts of an expression.
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For nested parentheses, work from the innermost set to the outermost.
- Example:
2 + (3 * (4 - 1))First calculate(4 - 1), then3 * 3and lastly add 2.
- Example:
2. Exponents
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Once parentheses are handled, exponents (powers and roots) are evaluated.
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This includes squaring, cubing, square roots, and other power operations.
- Example:
3^2 + 4calculates3^2, and then adds4.
- Example:
3. MD Multiplication and Division
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Multiplication and division are performed next, in the order they appear from left to right.
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They share the same priority level.
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It's vital to work from left to right if both operations are present.
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Example:
10 / 2 * 5is calculated from left to right,10/2is done before*5.
4. AS Addition and Subtraction
- Finally, addition and subtraction are performed, again from left to right.
- They also share the same level of priority.
- Example:
8 - 2 + 3is solved left to right, i.e.,(8-2)+3.
Practical Application
| Order | Operation | Example | Explanation |
|---|---|---|---|
| 1 | Parentheses | (5+2) * 3 |
5 + 2 is calculated first because it is inside parenthesis resulting in 7*3 |
| 2 | Exponents | 2^3 + 4 |
2^3 (which is 8) is calculated before the addition to 4. |
| 3 | Multiplication/Division | 10 / 2 * 5 |
10 / 2 calculated before *5. |
| 4 | Addition/Subtraction | 10 - 4 + 2 |
10 - 4 is performed first, followed by addition to 2. |
- Following this sequence ensures consistent results, regardless of who is solving the problem.
- Without this standard, mathematical expressions could be interpreted differently, leading to incorrect results.
- It applies to simple arithmetic, more complicated algebra, and other areas of mathematics.
By consistently following the order of operations, using PE(MD)(AS) as a mnemonic, the expression will be solved correctly. This ensures that regardless of who or where the expression is being solved, the result will be consistent.
