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Simplification is the process of reducing a mathematical expression to its simplest form. This means performing all possible calculations and operations step-by-step to arrive at a single number or a simpler expression. Simplification is important because it helps us solve problems quickly and accurately, especially when dealing with complex expressions involving multiple operations like addition, subtraction, multiplication, division, powers, roots, and brackets.
Imagine you are shopping in a market in India and you want to quickly calculate the total cost of items with different prices and discounts. Simplifying the expression that represents the total cost helps you find the answer without confusion or mistakes.
To simplify expressions correctly, we must follow a special set of rules called the order of operations. These rules tell us the sequence in which to perform calculations so that everyone gets the same answer. Without these rules, the same expression could give different answers depending on how it is solved.
The order of operations is a set of rules that tells us the correct sequence to solve parts of a mathematical expression. The most common rule is called BODMAS or BIDMAS, which stands for:
Division and Multiplication have the same priority; we solve them from left to right. The same applies to Addition and Subtraction.
This means:
graph TD A[Start Simplification] --> B[Brackets] B --> C[Orders (Powers & Roots)] C --> D[Division and Multiplication (Left to Right)] D --> E[Addition and Subtraction (Left to Right)] E --> F[Final Answer]
Before simplifying expressions, let's quickly review the four basic arithmetic operations:
When simplifying expressions, remember:
For example, in the expression \( -3 \times 4 \), the result is \(-12\) because the negative sign affects the multiplication.
Brackets (also called parentheses) are used to group parts of an expression that should be solved first. Sometimes, brackets are nested inside other brackets, like this:
\( 5 - [3 - \{2 + (4 - 6)\}] + 7 \)
To simplify such expressions, always start with the innermost bracket and work your way outward.
By following this inside-out approach, you avoid mistakes and simplify expressions correctly.
Factorization means breaking down a number or expression into smaller parts called factors that multiply together to give the original number. For example, the factors of 24 are 2, 3, 4, 6, 8, and 12 because 2 x 12 = 24, 3 x 8 = 24, and so on.
When simplifying fractions or complex expressions, factorization helps us cancel common factors from the numerator (top part) and denominator (bottom part) to make calculations easier.
| Original Expression | After Factorization | After Cancellation |
|---|---|---|
| \( \frac{24 \times 15}{18 \times 10} \) | \( \frac{(2 \times 2 \times 2 \times 3) \times (3 \times 5)}{(2 \times 3 \times 3) \times (2 \times 5)} \) | \( \frac{2}{3} \) |
| \( \frac{36}{48} \) | \( \frac{(2 \times 2 \times 3 \times 3)}{(2 \times 2 \times 2 \times 3)} \) | \( \frac{3}{4} \) |
Notice how common factors like 2 and 5 are canceled out, simplifying the expression significantly.
Step 1: Solve inside the brackets first: \(5 + 4 = 9\).
Step 2: Replace the bracket with 9: \(3 + 6 \times 9 \div 3 - 7\).
Step 3: Perform multiplication and division from left to right:
First, \(6 \times 9 = 54\).
Then, \(54 \div 3 = 18\).
Expression becomes: \(3 + 18 - 7\).
Step 4: Perform addition and subtraction from left to right:
\(3 + 18 = 21\), then \(21 - 7 = 14\).
Answer: \(14\)
Step 1: Simplify the innermost bracket: \(4 - 6 = -2\).
Expression becomes: \(5 - [3 - \{2 + (-2)\}] + 7\).
Step 2: Simplify inside the curly braces: \(2 + (-2) = 0\).
Expression becomes: \(5 - [3 - 0] + 7\).
Step 3: Simplify inside the square brackets: \(3 - 0 = 3\).
Expression becomes: \(5 - 3 + 7\).
Step 4: Perform subtraction and addition from left to right:
\(5 - 3 = 2\), then \(2 + 7 = 9\).
Answer: \(9\)
Step 1: Factorize each number:
Step 2: Write the fraction with factors:
\(\frac{(2 \times 2 \times 2 \times 3) \times (3 \times 5)}{(2 \times 3 \times 3) \times (2 \times 5)}\)
Step 3: Cancel common factors from numerator and denominator:
Remaining factors:
Numerator: \(2 \times 2 \times 3\)
Denominator: \(3 \times 2\)
Step 4: Simplify remaining factors:
Numerator: \(2 \times 2 \times 3 = 12\)
Denominator: \(3 \times 2 = 6\)
Step 5: Simplify the fraction \( \frac{12}{6} = 2 \).
Answer: \(2\)
Step 1: Simplify inside the brackets: \(1.2 - 0.6 = 0.6\).
Expression becomes: \(0.75 + \frac{2}{5} \times 0.6\).
Step 2: Multiply \(\frac{2}{5}\) by \(0.6\):
\(\frac{2}{5} \times 0.6 = \frac{2}{5} \times \frac{6}{10} = \frac{12}{50} = 0.24\).
Step 3: Add \(0.75 + 0.24 = 0.99\).
Answer: \(0.99\)
Step 1: Simplify inside the brackets: \(2 + 1 = 3\).
Expression becomes: \(\sqrt{81} + 3^2 \times 3\).
Step 2: Calculate the square root: \(\sqrt{81} = 9\).
Step 3: Calculate the power: \(3^2 = 9\).
Expression becomes: \(9 + 9 \times 3\).
Step 4: Multiply: \(9 \times 3 = 27\).
Step 5: Add: \(9 + 27 = 36\).
Answer: \(36\)
When to use: When simplifying expressions with multiple nested brackets.
When to use: When expressions contain mixed fractions and decimals.
When to use: When simplifying large numerical fractions.
When to use: When expressions have both multiplication and division operations.
When to use: When expanding expressions with subtraction or negative numbers.
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