Powers, Exponents, Surds & Number System for GATE Aptitude 2027: Shortcuts & Solved PYQs

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Quick Summary: Powers, exponents, surds, logarithms and the number system contribute 1 to 3 marks in GATE General Aptitude 2027 across every paper (DA, RA, CS, ME, EE, CE, EC). The questions test a handful of laws: exponent rules, log identities, unit-digit cycles, divisibility, remainders and counting factors. This guide covers every rule GATE uses, 8 solved previous-year and GATE-style questions, a 5-question practice set, a one-page formula sheet and the traps that cost a mark you had already earned.

🎯 Score 15/15 in General Aptitude. Practise this topic with 100+ exam-level questions, detailed solutions and All-India ranking in the Best GATE Aptitude Test Series 2027 by PiyushAI. Topic-wise tests + full-length GA mocks, valid till GATE 2027.

📋 Table of Contents

Why this topic matters in GATE GA
What the official GA syllabus says
Laws of exponents
Surds and rationalisation
Logarithms
Number system: divisibility, unit digits, remainders, factors
Shortcuts and traps
Solved PYQs and GATE-style questions
Practice set
One-page formula sheet
Common mistakes
FAQs

Why this topic matters in GATE GA

This is the chapter where GATE rewards people who know rules and punishes people who compute. Nobody can evaluate 2171⁷ in the exam hall, but anybody who knows that unit digits of powers repeat in cycles of four can find its last digit in ten seconds. The same applies to logarithms: a question that looks like it needs a calculator usually collapses to one identity.

Across DA, RA, CS, ME, EE, CE and EC the GA section is common, so this chapter is worth the same to everybody. The questions come in two flavours: a 1-mark MCQ on an exponent or log identity, and a 2-mark NAT on unit digits, remainders or the number of factors. Both are quick if the rule is in your head and impossible if it is not.

Here is roughly how much this cluster has contributed. Aspirants who want 15/15 in GA should be able to finish every question in this chapter under 60 seconds.

Year	Marks from this topic (approximate, across papers)	Typical question type
2019	1 to 2	Exponent equation, number of integers in a range
2020	1 to 2	Logarithm identity, surd simplification
2021	1 to 3	Remainder, divisibility
2022	1 to 2	Exponent comparison, unit digit
2023	1 to 2	Powers of 2 in a sequence, factor counting
2024	1 to 3	Logarithm equation, HCF and LCM
2025	1 to 2	Exponential growth, last digit
2026	1 to 2	Surds, perfect squares in a range

What the official GA syllabus says

Official GATE GA syllabus, Quantitative Aptitude: “Numerical computation and estimation: ratios, percentages, powers, exponents and logarithms, permutations and combinations, and series.”

“Powers, exponents and logarithms” is the exact phrase. The number-system questions (divisibility, remainders, factors) come under “numerical computation and estimation”. The complete syllabus breakdown and weightage is in the GATE General Aptitude 2027 hub. The previous post in this series covered Ratio, Proportion, Percentage and Averages.

Laws of exponents

For a positive base a and any real m, n:

a^m × aⁿ = a^m+n; a^m ÷ aⁿ = a^m−n
(a^m)ⁿ = a^mn; (ab)ⁿ = aⁿbⁿ; (a/b)ⁿ = aⁿ/bⁿ
a⁰ = 1 (a ≠ 0); a⁻ⁿ = 1/aⁿ; a^1/n = ⁿ√a; a^m/n = (ⁿ√a)^m
If a^m = aⁿ with a ≠ 0, 1, −1, then m = n. If aⁿ = bⁿ with n odd, then a = b.

The most common GATE trick is to write both sides with the same base. 4^x = 8^x−1 becomes 2^2x = 2^3x−3, so 2x = 3x − 3 and x = 3.

Comparing powers: to compare 2³⁰⁰, 3²⁰⁰ and 5¹⁰⁰, take the 100th root: 2³ = 8, 3² = 9, 5¹ = 5. So 3²⁰⁰ is largest, 5¹⁰⁰ is smallest.

✅ Solved Example (GATE 2023 style): If 3^x+2 − 3^x = 72, find x.
Solution: 3^x(3² − 1) = 72, so 3^x × 8 = 72 and 3^x = 9. Hence x = 2. (Check: 3⁴ − 3² = 81 − 9 = 72.)

✅ Solved Example (GATE 2023 style): A bacterial culture doubles every 3 hours. If it starts with 500 cells, how many cells are there after 24 hours?
Solution: Number of doublings = 24/3 = 8. Cells = 500 × 2⁸ = 500 × 256 = 1,28,000.

Surds and rationalisation

A surd is an irrational root such as √2 or ³√5. Rules:

√a × √b = √(ab); √a / √b = √(a/b); but √a + √b ≠ √(a + b).
Rationalise 1/(√a + √b) by multiplying numerator and denominator by (√a − √b), giving (√a − √b)/(a − b).
Nested surds: √(a + 2√b) = √x + √y where x + y = a and xy = b. For √(7 + 2√12): x + y = 7, xy = 12, so x = 4, y = 3, and the answer is 2 + √3.
Useful values: √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236, √10 ≈ 3.162.

✅ Solved Example (GATE 2023 style): If x = 1/(√5 − 2), what is x + 1/x?
Solution: Rationalise: x = (√5 + 2)/(5 − 4) = √5 + 2. Then 1/x = √5 − 2. Sum = 2√5, approximately 4.472.

Logarithms

log_ax = y means a^y = x, with a > 0, a ≠ 1, x > 0. Everything else follows from the exponent laws.

log(mn) = log m + log n; log(m/n) = log m − log n; log m^k = k log m
log_aa = 1; log_a1 = 0; a^log_ax = x
Change of base: log_ax = log_bx / log_ba; in particular log_ab × log_ba = 1
log_a^kx = (1/k) log_ax
Values to remember: log₁₀2 ≈ 0.3010, log₁₀3 ≈ 0.4771, log₁₀5 = 1 − log₁₀2 ≈ 0.6990, log₁₀7 ≈ 0.8451

Number of digits: an integer N has floor(log₁₀N) + 1 digits. So 2¹⁰⁰ has floor(100 × 0.3010) + 1 = 30 + 1 = 31 digits.

✅ Solved Example (GATE 2023 style): If log₂x + log₄x = 6, find x.
Solution: log₄x = (1/2) log₂x, so (3/2) log₂x = 6 and log₂x = 4. Hence x = 16. (Check: log₂16 = 4, log₄16 = 2, sum 6.)

Number system: divisibility, unit digits, remainders, factors

Divisibility rules

By 2, 5, 10: look at the last digit. By 4: last two digits divisible by 4. By 8: last three digits divisible by 8.
By 3 and 9: digit sum divisible by 3 or 9.
By 11: (sum of digits in odd places) − (sum of digits in even places) is 0 or a multiple of 11.
By 6: divisible by both 2 and 3. By 12: by both 3 and 4. By 7: double the last digit and subtract from the rest; repeat.

Unit digit of a power

Only the unit digit of the base matters. Digits 0, 1, 5, 6 never change. Digit 4 alternates 4, 6 (odd power gives 4, even gives 6); digit 9 alternates 9, 1. Digits 2, 3, 7, 8 repeat in cycles of four:

Unit digit of base	Power 1	Power 2	Power 3	Power 4
2	2	4	8	6
3	3	9	7	1
7	7	9	3	1
8	8	4	2	6

Divide the exponent by 4. Remainder 1, 2, 3 picks column 1, 2, 3; remainder 0 picks column 4. Unit digit of 7²⁰²⁷: 2027 = 4 × 506 + 3, so column 3, answer 3.

Remainders

Remainders respect addition and multiplication. To find 2¹⁰⁰ mod 7: 2³ = 8 ≡ 1 (mod 7), and 100 = 3 × 33 + 1, so 2¹⁰⁰ ≡ (2³)³³ × 2 ≡ 1 × 2 = 2. Find a power of the base that leaves remainder 1 (or −1) and the rest is arithmetic.

Factors, HCF and LCM

If N = p^a q^b r^c (prime factorisation), then the number of factors is (a + 1)(b + 1)(c + 1) and the sum of factors is (1 + p + … + p^a)(1 + q + … + q^b)(1 + r + … + r^c). For two numbers, HCF × LCM = product of the numbers.

✅ Solved Example (GATE 2023 style): How many factors does 360 have, and how many of them are odd?
Solution: 360 = 2³ × 3² × 5. Factors = (3 + 1)(2 + 1)(1 + 1) = 4 × 3 × 2 = 24. Odd factors use no 2: (2 + 1)(1 + 1) = 6 (they are 1, 3, 5, 9, 15, 45).

The GATE Aptitude Test Series has a dedicated topic test on exponents, logs and number system with 40 timed questions of exactly this type.

Shortcuts and traps

💡 GATE Tip: In any exponent equation, your first move is “same base”. 2, 4, 8, 16, 32 are all powers of 2; 3, 9, 27, 81 are powers of 3; 5, 25, 125 are powers of 5. Most GATE exponent questions dissolve after this step.

💡 GATE Tip: For the last digit of a sum of powers, find the last digit of each term separately, then add the last digits and take the unit digit of that sum.

💡 GATE Tip: When a question gives log P, log Q, log R in a chain like log P = (1/2) log Q = (1/3) log R, set each equal to k, write P = 10^k, Q = 10^2k, R = 10^3k, and test the options.

💡 GATE Tip: The number of perfect squares between 1 and N inclusive is floor(√N). Between 100 and 1000 there are 31 − 9 = 22 perfect squares (from 10² to 31²).

❌ Writing (a + b)² = a² + b². The middle term 2ab does not vanish.
❌ Writing √(a + b) = √a + √b or log(a + b) = log a + log b. Neither is true.
❌ Treating a^m × b^m as (ab)^2m. It is (ab)^m.
❌ Using exponent remainder 0 as “column 0” in the unit-digit table. Remainder 0 means the fourth column.
❌ Counting 1 and N themselves out of the factor count. The formula (a + 1)(b + 1) includes both.
❌ Forgetting that log is undefined for zero and negative arguments; a “solution” that makes x ≤ 0 must be rejected.
❌ Confusing log_a(x²) = 2 log_ax with (log_ax)².

Solved previous-year and GATE-style questions

✅ Solved Example (GATE 2017, CS): The last digit of (2171)⁷ + (2172)⁹ + (2173)¹¹ + (2174)¹³ is: (A) 2 (B) 4 (C) 6 (D) 8
Solution: Only unit digits matter: 1⁷ ends in 1. 2⁹: 9 = 4 × 2 + 1, column 1, ends in 2. 3¹¹: 11 = 4 × 2 + 3, column 3, ends in 7. 4¹³: odd power, ends in 4. Sum of unit digits = 1 + 2 + 7 + 4 = 14. Answer: (B) 4.

✅ Solved Example (GATE 2011, CS): If log(P) = (1/2) log(Q) = (1/3) log(R), then which of the following is true? (A) Q² = PR (B) Q² = R²P (C) Q² = R³P (D) R = P²Q²
Solution: Set each equal to k: log P = k, log Q = 2k, log R = 3k. Then log(Q²) = 4k and log(PR) = k + 3k = 4k. Answer: (A) Q² = PR.

✅ Solved Example (GATE 2012, CS): Given that (1.001)¹²⁵⁹ = 3.52 and (1.001)²⁰⁶² = 7.85, the value of (1.001)³³²¹ is: (A) 2.23 (B) 4.33 (C) 11.37 (D) 27.64
Solution: 1259 + 2062 = 3321, so (1.001)³³²¹ = 3.52 × 7.85 = 27.632. Answer: (D) 27.64.

✅ Solved Example (GATE 2016, CS): If q^−a = 1/r, r^−b = 1/s and s^−c = 1/q, the value of abc is: (A) (rqs)⁻¹ (B) 0 (C) 1 (D) r + q + s
Solution: The three relations say q^a = r, r^b = s, s^c = q. Substitute: s = r^b = q^ab, and q = s^c = q^abc. Hence abc = 1. Answer: (C) 1.

✅ Solved Example (GATE 2015, CS): The number of divisors of 2100 is ____.
Solution: 2100 = 2² × 3 × 5² × 7. Number of divisors = (2 + 1)(1 + 1)(2 + 1)(1 + 1) = 3 × 2 × 3 × 2 = 36.

✅ Solved Example (GATE 2015): If log_x(5/7) = −1/3, then the value of x is:
Solution: x^−1/3 = 5/7, so x^1/3 = 7/5 and x = (7/5)³ = 343/125.

✅ Solved Example (GATE-style): What is the remainder when 3¹⁰⁰ is divided by 7?
Solution: 3⁶ = 729 = 7 × 104 + 1, so 3⁶ ≡ 1 (mod 7). 100 = 6 × 16 + 4, so 3¹⁰⁰ ≡ 3⁴ = 81 = 7 × 11 + 4. Remainder: 4.

✅ Solved Example (GATE-style): If the six-digit number 7×5423 is divisible by 9, the digit x is:
Solution: Digit sum = 7 + x + 5 + 4 + 2 + 3 = 21 + x. For divisibility by 9 the sum must be 27 (the only multiple of 9 reachable with a digit 0 to 9), so x = 6.

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Practice set

Attempt these in 6 minutes before opening the key.

If 2^x = 4^y = 8^z and xyz = 288, then x + y + z is (A) 22 (B) 24 (C) 11 (D) 36
The unit digit of 7⁹⁵ − 3⁵⁸ is (A) 0 (B) 4 (C) 6 (D) 2
If log₁₀2 = 0.3010, the number of digits in 5²⁰ is (A) 13 (B) 14 (C) 15 (D) 20
The HCF of two numbers is 12 and their LCM is 180. If one number is 36, the other is (A) 48 (B) 60 (C) 72 (D) 90
The value of √(11 + 2√30) is (A) √5 + √6 (B) √10 + 1 (C) √3 + 2√2 (D) 2 + √7

Show answer key

(A) x = 2y = 3z = 6t gives x = 6t, y = 3t, z = 2t; xyz = 36t³ = 288, t³ = 8, t = 2; sum = 11t = 22.
(B) 7⁹⁵: 95 = 4 × 23 + 3, unit digit 3. 3⁵⁸: 58 = 4 × 14 + 2, unit digit 9. 3 − 9 means borrow: 13 − 9 = 4.
(B) log 5²⁰ = 20(1 − 0.3010) = 13.98; digits = 13 + 1 = 14.
(B) Other number = HCF × LCM / 36 = 12 × 180/36 = 60.
(A) x + y = 11, xy = 30, so x = 6, y = 5: √6 + √5. Check: 6 + 5 + 2√30 = 11 + 2√30.

One-page formula sheet

1. a^maⁿ = a^m+n; a^m/aⁿ = a^m−n; (a^m)ⁿ = a^mn; a⁻ⁿ = 1/aⁿ; a⁰ = 1.
2. a^m/n = ⁿ√(a^m). Compare powers by taking a common root.
3. √a√b = √(ab). 1/(√a + √b) = (√a − √b)/(a − b).
4. √(a + 2√b) = √x + √y with x + y = a, xy = b.
5. log(mn) = log m + log n; log(m/n) = log m − log n; log m^k = k log m.
6. log_ax = log x / log a; log_ab × log_ba = 1; a^log_ax = x.
7. log₁₀2 = 0.3010, log₁₀3 = 0.4771, log₁₀5 = 0.6990, log₁₀7 = 0.8451. Digits of N = floor(log₁₀N) + 1.
8. Unit-digit cycles: 2, 3, 7, 8 repeat every 4; 4 and 9 repeat every 2; 0, 1, 5, 6 are constant.
9. Divisibility: 3 and 9 by digit sum; 4 by last two digits; 8 by last three; 11 by alternating sum.
10. Factors of p^aq^br^c = (a + 1)(b + 1)(c + 1). HCF × LCM = product of two numbers.
11. Remainders: find a power of the base that is ≡ 1 or −1, then reduce the exponent.
12. Perfect squares up to N: floor(√N). Perfect cubes up to N: floor(³√N).

Common mistakes

❌ Multiplying exponents when bases are multiplied: a^m × aⁿ is a^m+n, not a^mn.
❌ Solving a^x = b^x as a = b without checking x = 0.
❌ Taking log of both sides of an equation that has a sum inside the log.
❌ Reading “number of digits” as log N rather than floor(log N) + 1.
❌ Forgetting to borrow when subtracting unit digits (3 − 9 gives 4, not −6).
❌ Using the divisibility rule for 11 with the wrong sign order and then not taking the absolute value.

FAQs

How many questions from powers, logs and number system come in GATE GA?

Usually 1 to 2 questions worth 1 to 3 marks per paper. They are quick, rule-based questions; the mark is lost only by not knowing the rule or by careless arithmetic.

Do I need to memorise log values?

Only log₁₀2 = 0.3010 and log₁₀3 = 0.4771. Every other common log value (5, 6, 8, 9, 12) follows from these two. GATE sometimes gives the values in the question anyway.

Is a scientific calculator available for these questions?

Yes, the GATE interface has a virtual scientific calculator. But using it for unit-digit or remainder questions is slower than the rule, and for a number like 2171⁷ it will overflow or round. Learn the rules; use the calculator only for the final multiplication.

Is the PiyushAI Aptitude Test Series enough for GA?

Yes. The Best GATE Aptitude Test Series 2027 has a topic test on exponents, logarithms and number system, tests on every other GA chapter, full-length GA mocks in the GATE interface, solved PYQs from 2010 to 2026 and All-India rank analysis. With the free guides on this site, it covers everything the GA section asks.

Are surds really asked in GATE?

Rarely as a standalone question, but surds appear inside geometry answers (√2, √3) and inside exponent questions written with fractional powers. Knowing how to rationalise and simplify nested surds is a five-minute investment that keeps paying.

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