Concept 1
A square root reverses squaring
For a non-negative real number n, √n is the non-negative number whose square is n. Thus √81=9 because 9²=81. The equation x²=81, however, has two solutions: x=9 and x=−9.
Important: √81 means the principal root 9; ± belongs to solving an equation such as x²=81.
Concept 2
Recognise perfect squares quickly
| n | 1–5 | 6–10 | 11–15 | 16–20 |
|---|---|---|---|---|
| n² | 1, 4, 9, 16, 25 | 36, 49, 64, 81, 100 | 121, 144, 169, 196, 225 | 256, 289, 324, 361, 400 |
A perfect square has even exponents in its prime factorisation. It can end in 0, 1, 4, 5, 6 or 9, but never in 2, 3, 7 or 8.
Concept 3
Take one factor from every equal pair
Since 1764=2²×3²×7², √1764=2×3×7=42. For a non-perfect square, move complete pairs outside: √72=√(2³×3²)=6√2.
Concept 4
Bracket a root before estimating
Because 12²=144 and 13²=169, √150 lies between 12 and 13 and is nearer 12. Near a², √(a²+d)≈a+d/(2a) when d is small; therefore √145≈12+1/24≈12.042.
Sanity check: square your estimate mentally. √150 cannot be 15 because 15²=225.
Concept 5
A surd preserves an irrational root exactly
√2, 3√5 and ∛7 are surds because their roots are irrational. √9 is not a surd because it equals 3. Simplest form has no removable perfect-power factor inside the radical.
Product law
For a,b≥0, √a·√b=√(ab). Thus √3·√12=6.
Quotient law
For a≥0 and b>0, √a/√b=√(a/b).
Concept 6
Combine only like surds
Simplify first, then collect matching radical parts: 2√8+3√18=4√2+9√2=13√2. Unlike surds such as √2 and √3 cannot be merged by addition.
Distribute when multiplying. Conjugates are useful because (√3+√2)(√3−√2)=3−2=1.
Concept 7
Remove surds from a denominator
For one surd, multiply top and bottom by it: 5/√3=5√3/3. For two terms, use the conjugate: 1/(√5+2)=(√5−2)/(5−4)=√5−2.
Conjugate rule: (a+b)(a−b)=a²−b² creates a rational denominator.
Worked methods
Examples with reasoning
Find √2304
2304=2⁸×3², so √2304=2⁴×3=48.
Simplify √192
192=64×3, so √192=8√3.
Simplify 5√12−2√27
10√3−6√3=4√3.
Rationalise 3/(2√5)
Multiply by √5/√5 to get 3√5/10.
Evaluate (√7+√3)²
7+3+2√21=10+2√21.
Exam traps
Errors to catch before they cost marks
False addition law
√(a+b) is generally not √a+√b; √25=5, not 7.
Missing absolute value
For real x, √(x²)=|x|, not always x.
Unlike surds
2√3+5√2 cannot become 7√5.
Negative radicands
The real-number product law requires both radicands to be non-negative.
Stopping early
√48 is not simplest form; it equals 4√3.
Wrong conjugate
The conjugate of a+√b is a−√b.
Original practice
Check understanding after learning
Original syllabus-aligned questions—not claimed previous-year questions.
Rapid revision
The one-minute sheet
- √n is the non-negative principal root; x²=n may have positive and negative solutions.
- A perfect square has even prime-factor exponents.
- Extract complete square factors: √(a²b)=a√b for a≥0.
- Combine only like surds after simplifying.
- Use a conjugate for a two-term surd denominator.
- Check estimates against neighbouring perfect squares.
Source ledger
Official foundations
- SSC CGL official syllabus: Mathematical Abilities
- NCERT official mathematics exemplar directory
- NCERT official textbook portal
Explanations, worked examples and practice are original. Official references are credited and not reproduced. Drafted 24 September 2026.
Averages: weighted means, combined groups and exam shortcuts.