Simplification — Study Notes

Overview

Simplification is one of the most scoring yet time-sensitive topics in the SSC GD Elementary Mathematics section. It typically accounts for 3–5 questions in the exam, testing your ability to reduce complex numerical expressions to a single value quickly and accurately. The core skill is applying the BODMAS rule (order of operations) systematically while handling fractions, decimals, surds, and roots.

Mastering simplification is essential because it forms the foundation for almost every other mathematics topic—be it percentage, profit-loss, or time-distance. Questions often appear deceptively simple but contain traps like misplaced brackets or mixed operations. A solid grip on BODMAS, comfort with square and cube roots, and fluency in surd manipulation will ensure you never drop marks here. Most candidates lose time by solving left-to-right without structure; disciplined application of rules is what separates high scorers.

In the exam, expect a mix of straightforward BODMAS questions, surd simplification problems, and questions combining roots with basic operations. Speed matters—aim to solve each simplification question in under 90 seconds.

Key Concepts

• **BODMAS Rule**: Operations must be performed in strict order—Brackets first, then Orders (powers/roots), then Division and Multiplication (left to right), finally Addition and Subtraction (left to right). Never work left-to-right blindly.

• **Brackets hierarchy**: When nested brackets appear, solve innermost first: ( ) then { } then [ ]. Always clear all brackets before moving to other operations.

• **Division and Multiplication are equal priority**: These two operations have the same precedence. Perform them as they appear from left to right, not all division first or all multiplication first.

• **Addition and Subtraction are equal priority**: Similarly, addition and subtraction share equal status. Execute them left to right after all higher operations are complete.

• **Surds**: A surd is an irrational root that cannot be simplified to a rational number, written as √n. Operations on surds follow algebraic rules: √a × √b = √(ab) and √a / √b = √(a/b).

• **Rationalizing the denominator**: When a surd appears in the denominator, multiply both numerator and denominator by the surd to eliminate it. For expressions like 1/(a + √b), multiply by the conjugate (a − √b).

• **Perfect squares and cubes**: Memorize squares up to 30 and cubes up to 20. This allows instant recognition: √625 = 25, ∛1728 = 12, saving crucial seconds during exams.

• **Approximation for non-perfect roots**: For roots like √50 or √80, factor out perfect squares: √50 = √(25×2) = 5√2. This simplifies calculation and comparison tasks.

Formulas / Key Facts

• **BODMAS order**: Brackets → Orders (√, ², ³) → Division → Multiplication → Addition → Subtraction (same-level operations go left to right)

• **Key square roots**: √1=1, √4=2, √9=3, √16=4, √25=5, √36=6, √49=7, √64=8, √81=9, √100=10, √121=11, √144=12, √169=13, √196=14, √225=15, √256=16, √289=17, √324=18, √361=19, √400=20, √441=21, √484=22, √529=23, √576=24, √625=25

• **Key cube roots**: ∛1=1, ∛8=2, ∛27=3, ∛64=4, ∛125=5, ∛216=6, ∛343=7, ∛512=8, ∛729=9, ∛1000=10

• **Surd multiplication**: √a × √b = √(ab). Example: √3 × √12 = √36 = 6

• **Surd division**: √a / √b = √(a/b). Example: √50 / √2 = √25 = 5

• **Rationalizing 1/√a**: Multiply by √a/√a to get √a/a

• **Conjugate rationalization**: 1/(a + √b) = (a − √b)/[(a + √b)(a − √b)] = (a − √b)/(a² − b)

• **Fraction operations in BODMAS**: Treat fractions as division. Simplify inside brackets first, then apply BODMAS to the entire expression.

Worked Examples

**Example 1: Basic BODMAS** Simplify: 48 ÷ 8 + 5 × 3 − 12

*Solution:* Step 1: No brackets or orders. Start with Division and Multiplication (left to right). 48 ÷ 8 = 6 5 × 3 = 15

Step 2: Rewrite: 6 + 15 − 12

Step 3: Addition and Subtraction (left to right): 6 + 15 = 21 21 − 12 = 9

**Answer: 9**

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**Example 2: Nested Brackets** Simplify: 120 − [35 − {28 − (15 − 7)}]

*Solution:* Step 1: Solve innermost bracket first: (15 − 7) = 8 Expression becomes: 120 − [35 − {28 − 8}]

Step 2: Solve curly bracket: {28 − 8} = 20 Expression becomes: 120 − [35 − 20]

Step 3: Solve square bracket: [35 − 20] = 15 Expression becomes: 120 − 15

Step 4: Final subtraction: 120 − 15 = 105

**Answer: 105**

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**Example 3: Surd Simplification** Simplify: √48 + √75 − √12

*Solution:* Step 1: Factor out perfect squares: √48 = √(16×3) = 4√3 √75 = √(25×3) = 5√3 √12 = √(4×3) = 2√3

Step 2: Combine like surds: 4√3 + 5√3 − 2√3 = (4 + 5 − 2)√3 = 7√3

**Answer: 7√3**

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**Example 4: Rationalization** Rationalize: 1/(3 + √2)

*Solution:* Step 1: Multiply numerator and denominator by conjugate (3 − √2): = 1×(3 − √2) / [(3 + √2)(3 − √2)]

Step 2: Apply (a+b)(a−b) = a² − b²: = (3 − √2) / (9 − 2) = (3 − √2) / 7

**Answer: (3 − √2)/7**

Common Mistakes

• **Left-to-right calculation without BODMAS**: Students often compute 5 + 3 × 2 as (5+3)×2 = 16 instead of 5 + (3×2) = 11. → Always apply BODMAS: Multiplication before addition gives 5 + 6 = 11.

• **Treating Division/Multiplication or Addition/Subtraction as separate passes**: Performing all multiplications first, then all divisions creates errors. → Division and Multiplication have equal priority—solve left to right. Same applies to Addition and Subtraction.

• **Forgetting to clear innermost brackets first**: Jumping to outer brackets leads to wrong results. → Always solve ( ) before { } before [ ]. Work inside-out systematically.

• **Incorrectly combining unlike surds**: Adding √2 + √3 as √5 is wrong. → You can only combine surds with the same radicand: 2√5 + 3√5 = 5√5, but √2 and √3 remain separate.

• **Leaving surds in the denominator**: Writing answers as 5/√3 loses marks in some exams. → Always rationalize: multiply by √3/√3 to get 5√3/3.

• **Calculation errors with negative signs**: 20 − (5 − 8) computed as 20 − 5 − 8 = 7 is wrong. → The bracket flips signs: 20 − (−3) = 20 + 3 = 23.

Quick Reference

• **BODMAS order**: Brackets → Orders → Div/Mult (left-right) → Add/Sub (left-right)
• **Surd multiplication**: √a × √b = √(ab); √3 × √12 = √36 = 6
• **Surd division**: √a / √b = √(a/b); √50 / √2 = √25 = 5
• **Rationalize 1/√a**: Multiply by √a/√a to eliminate surd in denominator
• **Memorize squares 1–25 and cubes 1–12**: Instant recognition saves 10–15 seconds per question
• **Nested brackets solve inside-out**: ( ) first, then { }, then [ ]