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Simplification is a fundamental skill in quantitative aptitude that involves reducing complex numerical expressions into simpler, more manageable forms. This skill is especially important in competitive exams, where time is limited and accuracy is crucial. By mastering simplification, you can solve problems faster, avoid errors, and gain confidence in handling a variety of numerical challenges.
Whether you are dealing with integers, fractions, decimals, surds, or algebraic expressions, the goal remains the same: to make the expression easier to work with without changing its value. This chapter will guide you through the essential concepts, rules, and techniques of simplification, starting from the basics and progressing to more advanced topics.
When simplifying any mathematical expression, it is important to perform operations in the correct order. The acronym BODMAS helps us remember this order:
Following BODMAS ensures that everyone simplifies expressions consistently and correctly.
graph TD A[Start] --> 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[Result]
Without following the correct order, you might get different answers for the same expression. For example, consider the expression:
\( 8 + 2 \times 5 \)
If you add first, \(8 + 2 = 10\), then multiply by 5 to get 50. But following BODMAS, multiply first: \(2 \times 5 = 10\), then add 8 to get 18, which is the correct answer.
Fractions represent parts of a whole and are written as \(\frac{a}{b}\), where \(a\) is the numerator and \(b\) is the denominator. Decimals are another way to represent fractions, especially those with denominators that are powers of 10.
To simplify fractions:
This reduces the fraction to its simplest form.
Sometimes, converting a fraction to a decimal makes calculations easier, especially if the decimal is terminating (ends after a few digits). However, if the decimal is recurring (repeats infinitely), it is better to work with the fraction directly to avoid rounding errors.
| Fraction | Steps to Simplify | Decimal Equivalent | Remarks |
|---|---|---|---|
| \(\frac{18}{24}\) | HCF of 18 and 24 is 6. Divide numerator and denominator by 6. \(\frac{18 \div 6}{24 \div 6} = \frac{3}{4}\) | 0.75 | Decimal is terminating and easy to use. |
| \(\frac{5}{8}\) | Already in simplest form. | 0.625 | Decimal is terminating. |
| \(\frac{7}{12}\) | HCF is 1 (already simplest). | 0.5833... (recurring) | Better to keep as fraction for accuracy. |
Surds are irrational roots, such as \(\sqrt{2}\), that cannot be simplified into exact decimals. Indices (or exponents) denote repeated multiplication, such as \(a^n\) meaning \(a\) multiplied by itself \(n\) times.
Understanding the rules for surds and indices helps simplify expressions efficiently.
Rationalizing the denominator means eliminating surds from the denominator of a fraction by multiplying numerator and denominator by the surd.
For example:
\[\frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{\sqrt{a} \times \sqrt{a}} = \frac{\sqrt{a}}{a}\]
Step 1: Simplify inside the brackets first (B in BODMAS): \(2 + 3 = 5\)
Step 2: Now the expression is \(5 \times 5 - 4^2\)
Step 3: Evaluate the power (O in BODMAS): \(4^2 = 16\)
Step 4: Multiply (M in BODMAS): \(5 \times 5 = 25\)
Step 5: Subtract (S in BODMAS): \(25 - 16 = 9\)
Answer: 9
Step 1: Simplify \(\frac{18}{24}\) by dividing numerator and denominator by their HCF, 6:
\(\frac{18 \div 6}{24 \div 6} = \frac{3}{4}\)
Step 2: Now the expression is \(\frac{3}{4} + \frac{5}{8}\)
Step 3: Find the LCM of denominators 4 and 8, which is 8.
Step 4: Convert \(\frac{3}{4}\) to \(\frac{6}{8}\) to have a common denominator.
Step 5: Add the fractions: \(\frac{6}{8} + \frac{5}{8} = \frac{11}{8}\)
Answer: \(\frac{11}{8}\) or \(1 \frac{3}{8}\)
Step 1: Simplify \(\sqrt{50}\) by expressing 50 as \(25 \times 2\):
\(\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}\)
Step 2: Now the expression is \(5\sqrt{2} + 3\sqrt{2}\)
Step 3: Since both terms have \(\sqrt{2}\), add the coefficients:
\(5 + 3 = 8\)
Step 4: Final simplified expression is \(8\sqrt{2}\)
Answer: \(8\sqrt{2}\)
Step 1: Multiply the powers with the same base \(a\) by adding exponents:
\(a^5 \times a^3 = a^{5+3} = a^8\)
Step 2: Now the expression is \(\frac{a^8}{a^4}\)
Step 3: Divide powers with the same base by subtracting exponents:
\(a^{8-4} = a^4\)
Answer: \(a^4\)
Step 1: Factor numerators and denominators:
12 = \(2^2 \times 3\), 15 = \(3 \times 5\), 10 = \(2 \times 5\), 9 = \(3^2\)
Step 2: Write the expression with factors:
\(\frac{2^2 \times 3}{3 \times 5} \times \frac{2 \times 5}{3^2}\)
Step 3: Cancel common factors:
Cancel one 3 in numerator and denominator, cancel 5 in numerator and denominator.
Expression becomes \(\frac{2^2}{1} \times \frac{2}{3}\)
Step 4: Multiply remaining terms:
\(2^2 \times 2 = 4 \times 2 = 8\)
Denominator is 3
Step 5: Final simplified fraction is \(\frac{8}{3}\)
Answer: \(\frac{8}{3}\) or \(2 \frac{2}{3}\)
When to use: When dealing with expressions containing multiple operations.
When to use: When fractions are complex but decimals are terminating and simple.
When to use: When simplifying fractions or expressions involving multiples.
When to use: When expressions involve square roots or cube roots.
When to use: When expressions have multiple powers with the same base.
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