Surds and Indices
Surds and Indices
Surds and indices questions in NTPC are pure rule application: laws of exponents, fractional powers, rationalisation and square roots of decimals. They appear in all three cycles (fractional powers in April 2016, surd simplification in December 2020, index laws in August 2025, an exponent equation in the March 2026 Graduate CBT-1), so learn every law in the table below until you can apply it without writing.
1. Laws of Indices (a, b > 0, m, n rational)
| Law | Form | Example |
|---|---|---|
| Product | aᵐ × aⁿ = a^(m + n) | 2³ × 2⁴ = 2⁷ = 128 |
| Quotient | aᵐ ÷ aⁿ = a^(m − n) | 5⁷ ÷ 5⁵ = 25 |
| Power of power | (aᵐ)ⁿ = a^(mn) | (3²)³ = 3⁶ = 729 |
| Power of product | (ab)ⁿ = aⁿbⁿ | (2 × 5)³ = 1,000 |
| Power of quotient | (a/b)ⁿ = aⁿ/bⁿ | (2/3)³ = 8/27 |
| Zero power | a⁰ = 1 | 1,999⁰ = 1 |
| Negative power | a^(−n) = 1/aⁿ, and (a/b)^(−n) = (b/a)ⁿ | (2/5)^(−2) = 25/4 |
| Fractional power | a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)ᵐ | 27^(2/3) = 3² = 9 |
| Equal bases | aˣ = aʸ means x = y (a > 0, a ≠ 1) | 2^(x+1) = 2⁵ gives x = 4 |
Fractional power in 5 seconds: take the root first, then the power. 64^(−2/3): cube root 4, square 16, negative gives 1/16.
2. Surd Rules
| Rule | Form | Example |
|---|---|---|
| Product | √a × √b = √(ab) | √8 × √2 = 4 |
| Quotient | √a ÷ √b = √(a/b) | √75 ÷ √3 = 5 |
| Simplest form | Pull out perfect squares | √98 = 7√2, √180 = 6√5 |
| Like surds only add | a√k + b√k = (a + b)√k | 3√5 + 2√5 = 5√5 |
| Rationalise a + √b | multiply by a − √b | 1/(3 + √2) = (3 − √2)/7 |
| Rationalise √a + √b | multiply by √a − √b | 1/(√5 + √3) = (√5 − √3)/2 |
| √(a + 2√b) | = √x + √y where x + y = a and xy = b | √(7 + 4√3) = √(7 + 2√12) = 2 + √3 |
✗ √a + √b = √(a + b) | ✓ √9 + √16 = 7, but √25 = 5; only products and quotients combine under one root
Comparing surds of different orders: raise all to the LCM of the root orders. For ∛4, √3 and ⁶√15 the LCM is 6: compare 4² = 16, 3³ = 27 and 15. So √3 is largest.
Square roots of decimals: pair the decimal places. √29.16 = 5.4 because 2,916 = 54² and 29.16 has 2 decimal places (1 pair). A decimal with an ODD number of decimal places (√0.4, √2.916) does not follow the pattern.
3. Standard Values
| √2 | √3 | √5 | √6 | √7 | √10 |
|---|---|---|---|---|---|
| 1.414 | 1.732 | 2.236 | 2.449 | 2.646 | 3.162 |
Infinite radicals (CBT-2): √(x + √(x + √(x + ...))) = (1 + √(1 + 4x))/2, so with x = n(n + 1) the value is n + 1 (√(12 + √(12 + ...)) = 4). And √(x√(x√(x...))) = x.
हिंदी नोट: घातांक के नियम में आधार समान होने पर ही घातें जोड़ी या घटाई जाती हैं। करणी (surd) में केवल समान करणी ही जोड़ी जा सकती हैं, जैसे 3√5 + 2√5 = 5√5।
Exam Pointer: Verified NTPC patterns from this block: index laws with the same base (UG CBT-1, August 2025), unknown in the exponent (Graduate CBT-1, March 2026), fractional and negative powers (April 2016, September 2025), surd simplification (December 2020), and rationalisation plus the "if √2916 = 54" decimal chain (2020-21 cycle). The trap options are adding unlike surds and ignoring the negative sign in a negative power.
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Laws of indices with the same base
[PYQ: NTPC UG CBT-1 8-Aug-2025 Shift-1]
EXAM LEVEL
Q. Find the value of 13^(−6) ÷ 13⁴ × 13¹².
Same base, so work only with powers: −6 − 4 + 12 = 2. Value = 13² = 169.
Answer: 169
EXAMATLAS LEVEL
Q. Simplify (2^(n+4) − 2 × 2ⁿ) ÷ (2 × 2^(n+3)).
Take 2ⁿ common: numerator = 2ⁿ(16 − 2) = 14 × 2ⁿ, denominator = 2ⁿ × 2⁴ = 16 × 2ⁿ. The 2ⁿ cancels and the value is 14/16 = 7/8, independent of n.
Answer: 7/8
Pattern 2: Unknown in the exponent
[PYQ: NTPC Graduate CBT-1 19-Mar-2026 Shift-3]
EXAM LEVEL
Q. If 3^(2x − 1) = 243, find x.
243 = 3⁵, so 2x − 1 = 5 and x = 3.
Answer: 3
EXAMATLAS LEVEL
Q. If 2ˣ = 3ʸ = 6^(−z), find 1/x + 1/y + 1/z.
Let each equal k. Then 2 = k^(1/x), 3 = k^(1/y) and 6 = k^(−1/z). Since 2 × 3 = 6, k^(1/x + 1/y) = k^(−1/z), so 1/x + 1/y = −1/z and the required sum is 0.
Answer: 0
Pattern 3: Fractional and negative powers
[PYQ: NTPC CBT-1 11-Apr-2016 Shift-2 | NTPC UG CBT-1 9-Sep-2025 Shift-3]
EXAM LEVEL
Q. Find the value of 64^(−2/3).
Cube root of 64 is 4, square is 16, the negative sign inverts it: 1/16.
Answer: 1/16
EXAMATLAS LEVEL
Q. Find the value of (0.008)^(−2/3) × (16/81)^(−3/4).
0.008 = (0.2)³, so (0.008)^(−2/3) = (0.2)^(−2) = 1/0.04 = 25. (16/81)^(−3/4) = (81/16)^(3/4); fourth root of 81/16 is 3/2 and its cube is 27/8. Product = 25 × 27/8 = 675/8 = 84.375.
Answer: 675/8 (84.375)
Pattern 4: Simplifying and adding surds
[PYQ: NTPC CBT-1 29-Dec-2020 Shift-1]
EXAM LEVEL
Q. Simplify √98 + √50 − √32.
√98 = 7√2, √50 = 5√2, √32 = 4√2. Total = (7 + 5 − 4)√2 = 8√2.
Answer: 8√2
EXAMATLAS LEVEL
Q. If √5 = 2.236, find √20 + √45 − √125 + √180.
Convert each to a multiple of √5: 2√5 + 3√5 − 5√5 + 6√5 = 6√5. Value = 6 × 2.236 = 13.416. Plugging 2.236 into each term separately wastes a minute and invites rounding errors.
Answer: 13.416
Pattern 5: Rationalising and symmetric expressions
[PYQ: NTPC 2020-21 cycle]
EXAM LEVEL
Q. Simplify (√5 + √3)/(√5 − √3).
Multiply top and bottom by (√5 + √3): numerator (√5 + √3)² = 8 + 2√15, denominator 5 − 3 = 2. Value = 4 + √15.
Answer: 4 + √15
EXAMATLAS LEVEL
Q. If x = (√5 + √3)/(√5 − √3) and y = (√5 − √3)/(√5 + √3), find x² + y².
x = 4 + √15 and y = 1/x = 4 − √15. So x + y = 8 and xy = 1. Then x² + y² = (x + y)² − 2xy = 64 − 2 = 62. Squaring each surd separately takes four times as long.
Answer: 62
Pattern 6: Square roots of decimals from a given root
[PYQ: NTPC 2020-21 cycle]
EXAM LEVEL
Q. If √3249 = 57, find √32.49 + √0.3249 + √0.003249.
Each decimal has an even number of decimal places, so the roots are 5.7, 0.57 and 0.057. Sum = 6.327.
Answer: 6.327
EXAMATLAS LEVEL
Q. If √1369 = 37, find √13.69 + √0.1369 + √0.001369 + √136900.
√13.69 = 3.7, √0.1369 = 0.37, √0.001369 = 0.037, and 136900 = 1369 × 100 so its root is 370. Sum = 3.7 + 0.37 + 0.037 + 370 = 374.107. The trap term is the last one, which moves the decimal the other way.
Answer: 374.107
Pattern 7: Comparing and ordering surds
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. Which is the largest: ∛4, √3 or ⁶√15?
LCM of 3, 2 and 6 is 6. Raise each to the 6th power: (∛4)⁶ = 16, (√3)⁶ = 27, (⁶√15)⁶ = 15. Largest is √3.
Answer: √3
EXAMATLAS LEVEL
Q. Arrange ∜6, ∛4 and √2 in ascending order.
LCM of 4, 3, 2 is 12. Twelfth powers: (∜6)¹² = 6³ = 216, (∛4)¹² = 4⁴ = 256, (√2)¹² = 2⁶ = 64. Ascending: √2, ∜6, ∛4.
Answer: √2 < ∜6 < ∛4
CBT-2 LEVEL
Pattern 8: Nested radicals and √(a + 2√b)
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. Find the value of √(12 + √(12 + √(12 + ...))).
Let the value be x. Then x = √(12 + x), so x² − x − 12 = 0, (x − 4)(x + 3) = 0 and x = 4 (it must be positive). Shortcut: 12 = 3 × 4, so the answer is the larger factor 4.
Answer: 4
EXAMATLAS LEVEL
Q. Find the value of √(7 + 4√3) − 1/(2 + √3).
7 + 4√3 = 7 + 2√12; find two numbers with sum 7 and product 12: 4 and 3. So √(7 + 4√3) = √4 + √3 = 2 + √3. Also 1/(2 + √3) = 2 − √3 after rationalising (denominator 4 − 3 = 1). Difference = (2 + √3) − (2 − √3) = 2√3.
Answer: 2√3
60-Second Revision
- Same base: add powers when multiplying, subtract when dividing, multiply for power of power.
- a^(m/n): root first, then power; a negative power flips the fraction.
- Only like surds add; simplify √98, √50 to multiples of √2 first.
- Rationalise with the conjugate; for x and 1/x use (x + y)² − 2xy.
- Decimal roots: even decimal places only; √(a + 2√b) = √x + √y with x + y = a, xy = b.
- Compare surds by raising to the LCM of root orders.
Next Step: Number System done. Now attempt the Number System questions in the ExamAtlas RRB NTPC 2026 mock tests and topic practice sets, and check every wrong answer against the matching pattern on this page.