Simplification — Study Notes for SSC MTS (Paper 1)

Overview

Simplification questions test your ability to perform arithmetic operations in the correct order and handle fractions, surds, and exponents accurately. In SSC MTS, expect 2–4 direct questions worth 8–16 marks. These are scoring questions if you follow rules systematically and avoid calculation errors.

The key skill is applying **BODMAS** (Brackets, Orders, Division/Multiplication, Addition/Subtraction) correctly while handling mixed operations involving whole numbers, fractions, decimals, surds (roots), and exponents (powers). Most errors happen from rushing through steps or misapplying operator priority. Master this topic early — it's the foundation for almost every other quantitative topic in the exam.

Success in simplification means: (1) knowing operation hierarchy cold, (2) quick fraction arithmetic, (3) confidence with basic surd rules and exponent laws, and (4) disciplined step-by-step calculation without mental shortcuts that introduce errors.

Key Concepts

• **BODMAS/BODMAS rule**: Operations must be performed in strict order — Brackets first, then Orders (exponents/roots), then Division and Multiplication (left to right), finally Addition and Subtraction (left to right). Ignoring this order guarantees wrong answers.

• **Brackets hierarchy**: Solve innermost brackets first. The nesting order is: ( ) inner parentheses, { } curly braces, [ ] square brackets. Work from inside out.

• **Fractions**: To add/subtract fractions, find the LCM of denominators and convert to equivalent fractions. For multiplication, multiply numerators and denominators directly. For division, multiply by the reciprocal of the divisor.

• **Surds (roots)**: √a × √b = √(ab); √a / √b = √(a/b); √a ± √b cannot be simplified unless a = b. Rationalization means removing surds from denominators by multiplying numerator and denominator by the conjugate or the surd itself.

• **Exponents (powers)**: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ / aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1 (a ≠ 0); a⁻ⁿ = 1/aⁿ; (ab)ⁿ = aⁿbⁿ. These laws apply universally and must be memorized cold.

• **Order of operations with mixed types**: When an expression has fractions, decimals, surds and exponents together, apply BODMAS strictly while respecting each number type's arithmetic rules. Convert everything to the same form (all fractions or all decimals) if it simplifies calculation.

• **Simplification strategy**: Break complex expressions into smaller chunks, solve brackets step by step, reduce fractions at every stage, and recheck your final answer by rough approximation.

Formulas / Key Facts

**BODMAS Priority Order:** 1. **B** — Brackets: ( ), { }, [ ] — innermost first 2. **O** — Orders: exponents (powers) and roots 3. **D/M** — Division and Multiplication (left to right, equal priority) 4. **A/S** — Addition and Subtraction (left to right, equal priority)

**Fraction Operations:**

• a/b + c/d = (ad + bc) / bd
• a/b − c/d = (ad − bc) / bd
• a/b × c/d = ac / bd
• a/b ÷ c/d = a/b × d/c = ad / bc

**Surd Rules:**

• √a × √b = √(ab)
• √a / √b = √(a/b)
• (√a)² = a
• Rationalization: 1/√a = √a/a; 1/(√a + √b) = (√a − √b)/(a − b)

**Exponent Laws:**

• aᵐ × aⁿ = aᵐ⁺ⁿ
• aᵐ / aⁿ = aᵐ⁻ⁿ
• (aᵐ)ⁿ = aᵐⁿ
• a⁰ = 1
• a⁻ⁿ = 1/aⁿ
• (ab)ⁿ = aⁿbⁿ
• (a/b)ⁿ = aⁿ/bⁿ

**Mixed Number:** a(b/c) = (ac + b)/c — convert to improper fraction before operations.

Worked Examples

**Example 1: BODMAS with mixed operations**

Simplify: 36 ÷ 4 + 5 × 3 − 8

**Solution:** Follow BODMAS strictly:

• No brackets, no orders
• Division and Multiplication first (left to right):
• 36 ÷ 4 = 9
• 5 × 3 = 15
• Expression becomes: 9 + 15 − 8
• Addition and Subtraction (left to right):
• 9 + 15 = 24
• 24 − 8 = 16

**Answer: 16**

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**Example 2: Simplification with fractions**

Simplify: 2/3 + 5/6 − 1/2

**Solution:** Find LCM of denominators 3, 6, 2 → LCM = 6 Convert each fraction:

• 2/3 = 4/6
• 5/6 = 5/6
• 1/2 = 3/6

Now: 4/6 + 5/6 − 3/6 = (4 + 5 − 3)/6 = 6/6 = 1

**Answer: 1**

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**Example 3: Exponents and roots**

Simplify: (2³ × 2⁵) / 2⁴ + √64

**Solution:** Handle exponents first:

• 2³ × 2⁵ = 2³⁺⁵ = 2⁸ = 256
• 256 / 2⁴ = 256 / 16 = 16

Handle root:

• √64 = 8

Final: 16 + 8 = 24

**Answer: 24**

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**Example 4: Complex brackets**

Simplify: 48 − [36 − {15 − (8 − 5)}]

**Solution:** Solve innermost bracket first:

• (8 − 5) = 3
• Expression: 48 − [36 − {15 − 3}]

Next bracket:

• 15 − 3 = 12
• Expression: 48 − [36 − 12]

Outer bracket:

• 36 − 12 = 24
• Expression: 48 − 24 = 24

**Answer: 24**

Common Mistakes

• **Mistake**: Adding/subtracting before multiplying/dividing when no brackets present.

**Fix**: Always complete all multiplication and division first (left to right), then do addition and subtraction (left to right). Write out intermediate steps to avoid this error.

• **Mistake**: Solving brackets in wrong order (outer before inner).

**Fix**: Always start with the innermost bracket and work outward. Mark each bracket type clearly as you solve.

• **Mistake**: Applying exponent laws incorrectly: writing 2³ × 3² = 6⁵.

**Fix**: Exponent addition rule (aᵐ × aⁿ = aᵐ⁺ⁿ) applies only when bases are identical. Here, bases differ (2 and 3), so compute each power separately: 8 × 9 = 72.

• **Mistake**: Adding surds incorrectly: √2 + √3 = √5.

**Fix**: You cannot add surds with different radicands. √2 + √3 stays as is. Only like surds can be combined: 2√3 + 5√3 = 7√3.

• **Mistake**: Forgetting to rationalize denominators containing surds in final answers.

**Fix**: SSC often expects rationalized form. If answer is 1/√2, multiply top and bottom by √2 to get √2/2.

Quick Reference

• **BODMAS order**: Brackets → Orders → Division/Multiplication (left to right) → Addition/Subtraction (left to right)
• **Fraction addition**: Find LCM of denominators first; for multiplication, no LCM needed — just multiply straight across
• **Exponent rules**: Same base → add/subtract powers; power of power → multiply exponents; a⁰ = 1; a⁻ⁿ = 1/aⁿ
• **Surd multiplication**: √a × √b = √(ab); cannot add √a + √b unless a = b
• **Rationalization**: Multiply by the surd (or conjugate) to clear the denominator
• **Always simplify step by step**: Don't try to do multiple operations mentally — write intermediate results to catch errors early