Power, Indices & Surds Tests — Pick a Level and Start
Tap a level to see every test in it. Start at Level 1 the day you finish the index laws, then move to Level 2 with the timer running to push accuracy under pressure.
Level 1 Basic Checking… View tests
One rule applied cleanly — combine powers of one base, kill a bracket with a zero index, turn 322/5 into 4, rationalise a single conjugate, split √(13 + 2√40), and evaluate an infinite nested radical. Attempt these without the clock while the rules are still fresh.
Level 2 Tier I Standard Checking… View tests
Real SSC CGL Tier I difficulty — two ideas chained in one question: recover the base from a fractional power, find the constant that makes an n-expression collapse, evaluate a nested radical and then rationalise using its value, close a power chain on x8 or 1/x, and count the indices that keep a shrinking power inside two bounds. This is the score that tells you the chapter is exam-ready.
Power, indices and surds is the fastest-scoring chapter in the SSC CGL Quantitative Aptitude paper. Almost every shift carries one or two questions that belong to it outright — an index equation to solve, a surd fraction to rationalise, a nested radical to evaluate, or four surds to put in order — and none of them need long calculation. They need the right rule applied in the right order.
These practice tests put twenty-five power, indices and surds questions in front of you in one sitting, which is what turns the index laws from something you can recite into something you apply without thinking. Every question comes with a full solution and the shortcut, in English or Hindi.
How Many Power, Indices and Surds Questions Are Asked in SSC CGL?
The chapter usually gets one to two direct questions, and its methods are reused inside simplification, algebra and the comparison-type questions that appear across the paper.
| Chapter | Tier I (out of 25) | Tier II Paper I (out of 30) |
|---|---|---|
| Power, Indices & Surds | 1–2 | 2–3 |
| Simplification | 1–2 | 2–3 |
| Number System | 2–3 | 3–4 |
| Square & Cube Roots | 1 | 1–2 |
| Algebra | 2–3 | 3–4 |
Typical spread across recent SSC CGL shifts. SSC does not publish a chapter-wise breakup, so treat these as planning ranges. Tier I maths is always 25 questions in a locked 15-minute section.
What You Will Practice in the Power, Indices & Surds Chapter
Core topics
- The eight index laws — product, quotient, power of a power, power of a product and quotient
- Zero index and negative index, including brackets that vanish and fractions that flip
- Fractional indices, roots of roots, and converting between root form and index form
- Decimal indices on composite bases such as 625, 216 and 4096
- Solving for an unknown index — one prime, two primes, roots on either side, unknown on both sides
- Shifted powers such as 3x+2 − 3x = 72, and repeated copies of one power
- Closed power chains and common-value questions of the ax = by = cz type
- Prime-factor exponent questions — writing a product as 2a3b5c and combining the indices
- Pure, mixed and like surds; taking factors in and out of a root; changing the order of a surd
- Rationalising monomial, binomial and three-term surd denominators, and telescoping chains
- Compound surds √(a ± 2√b) and their sums, differences and reciprocals
- Finite and infinite nested radicals, including the multiplicative form
- Comparing surds of different orders, fractional powers, large powers and power towers
- Approximation questions using given root values
The index laws you must know cold
| Law | Rule | Example |
|---|---|---|
| Product | am × an = am+n | 23 × 24 = 27 |
| Quotient | am ÷ an = am−n | 57 ÷ 53 = 54 |
| Power of a power | (am)n = amn | (32)4 = 38 |
| Power of a product | (ab)n = anbn | 63 = 23 × 33 |
| Zero index | a0 = 1 (a ≠ 0) | (37 − 211)0 = 1 |
| Negative index | a−n = 1/an | (2/5)−2 = 25/4 |
| Fractional index | ap/q = (q√a)p | 322/5 = 22 = 4 |
| Equal bases | am = an ⇒ m = n | a > 0, a ≠ 1 |
Multiply → add the indices, divide → subtract, bracket → multiply. Those three lines solve most of the chapter. Note that 0k = 0 for a positive k — only a non-zero base gives 1 at the zero index.
Surd rules and conversions
| Job | Rule | Example |
|---|---|---|
| Pure → mixed surd | n√(mna) = m·n√a | 3√108 = 33√4 |
| Mixed → pure surd | k·n√a = n√(kna) | 23√5 = 3√40 |
| Raise the order | n√a = nk√(ak) | 3√5 = 6√25 |
| Lower the order | nk√(ak) = n√a | 6√125 = √5 |
| Monomial denominator | a/√b = a√b / b | 4/√2 = 2√2 |
| Binomial denominator | 1/(p + q) = (p − q)/(p² − q²) | 1/(2 + √3) = 2 − √3 |
| Consecutive roots | 1/(√(n+1) + √n) = √(n+1) − √n | the telescoping identity |
When taking a number inside a root, raise it to the order of that root — inside a cube root the 2 becomes 8, not 4. That single slip accounts for most wrong answers in surd conversion questions.
Compound surds and nested radicals
- Splitting rule: √(a + 2√b) = √m + √n where m + n = a and mn = b. For the minus sign the answer is √m − √n.
- Make the 2 if it is missing: 4√5 = 2√20, so √(9 − 4√5) = √5 − 2. If there is no coefficient at all, double inside and divide by √2.
- Sum and difference: √(a+2√b) + √(a−2√b) = 2√m, and the difference is 2√n.
- Finite nests: work inside out and read the order written on each root sign.
- Infinite plus-nest: x² = a + x. When a = n(n+1) the value is n + 1, so √(30 + √(30 + …)) = 6.
- Infinite minus-nest: x² = a − x, giving n instead of n + 1 for the same a.
- Multiplicative nest: √(a√(a√a …)) = a itself, not the plus-nest value.
Comparing surds and powers
| Situation | Method | Example |
|---|---|---|
| Roots of different orders | Convert to the LCM order | √2, 3√3 → 6√8, 6√9 |
| Different fractional indices | Raise all to the LCM of the denominators | 31/2, 51/3 → 36 = 729, 54 = 625 |
| Large powers | Divide every index by their common factor | 236, 324, 518 → 64, 81, 125 |
| Sums with equal totals inside | Bigger product = bigger sum | √10 + √10 > √17 + √3 |
| Differences with equal gaps | Smaller numbers = bigger difference | √8 − √4 > √35 − √31 |
| Power towers | Read from the top down | 232 = 29 = 512, but (23)2 = 64 |
Also worth memorising
- Squares to 30, cubes to 15, and 1024 = 210, 3125 = 55, 4096 = 212 = 84 = 163
- √2 = 1.414, √3 = 1.732, √5 = 2.236, √6 = 2.449, √7 = 2.646, √10 = 3.162
- If x = a + √b and a² − b = 1, then 1/x = a − √b, so x + 1/x = 2a and x − 1/x = 2√b
- Two decimal places under a square root become one place outside; three places under a cube root become one
- Equal indices on different primes merge: 211 × 511 = 1011
- In a closed power chain, the product of all the indices equals the final index of the base
Solved Examples — SSC CGL Power, Indices & Surds
How the Power, Indices & Surds Chapter Tests Work
- 25 questions per test, with a full question palette — answered, skipped, marked for review, not visited.
- Optional 30-minute timer — with the clock to build exam speed, without it while you are still learning the rules.
- Marking: +2 correct, −0.5 wrong, 0 unattempted — the same scheme as the real paper.
- Solutions, shortcut tricks and concept notes open after you submit, question by question.
- English and हिंदी, chosen at the start of every test.
- Free with a TrickySSC account — sign in once with Google or your mobile number, and every test is open to you.
How to Prepare Power, Indices & Surds for SSC CGL
- Learn the eight index laws first, then never derive them again. Multiply → add, divide → subtract, bracket → multiply. Everything else in the chapter sits on top of these three.
- Memorise the perfect powers. Squares to 30, cubes to 15, and the fourth powers 16, 81, 256, 625, 1296, 2401, 4096. Recognising 4096 as 84 is often the whole question.
- Always reduce to one prime. Almost every unknown-index question becomes a one-line linear equation the moment 8, 27 and 125 are written as 23, 33 and 53.
- Practise the compound surd split until it is automatic. Find m and n with m + n = a and mn = b. Half the surd questions in the paper reduce to this one step.
- Never compare surds by eye. 3√3 and √2 differ by 0.03 — convert to a common order instead of estimating.
- Simplify before you substitute in approximation questions. Collect like surds or rationalise first, then use the given root value exactly once.
- Read what the question actually asks. Many items solve for x but ask for x², 2x or x − 5, and the un-finished value is always in the options.
- Do Level 1 untimed, then Level 2 with the clock. If a rationalisation takes more than twenty seconds, you are still deriving the rule instead of applying it.
- Study it with simplification and algebra. Do number system before it, and simplification and algebra right after — the same index and surd tricks run through all of them.
- Verify against a real paper. Once Level 2 is consistent, attempt a previous year paper and then a full mock test under locked sectional timing.
Power, Indices & Surds Study Notes
Every rule on this page, with worked examples and the ten traps that cost marks, is collected in the chapter notes. Read them once before you start Level 1.
SSC CGL घात, घातांक एवं करणी (Power, Indices & Surds) चैप्टर टेस्ट — हिंदी में
घात, घातांक एवं करणी SSC CGL गणित का सबसे तेज़ अंक देने वाला अध्याय है। लगभग हर शिफ्ट में इसके 1–2 प्रश्न सीधे आते हैं — जैसे घातांक वाला समीकरण हल करना, करणी वाली भिन्न का परिमेयीकरण, नेस्टेड मूल का मान निकालना, या चार करणियों को क्रम में लगाना — और इनमें से किसी में लंबी गणना नहीं चाहिए, बस सही नियम सही क्रम में लगाना है।
TrickySSC पर इस अध्याय के सभी टेस्ट हिंदी में उपलब्ध हैं — प्रश्न, विकल्प, विस्तृत हल और शॉर्टकट ट्रिक सब हिंदी में। दो स्तर हैं: लेवल 1 (आधारभूत) और लेवल 2 (टियर I स्तर)। प्रत्येक टेस्ट में 25 प्रश्न, वैकल्पिक 30 मिनट का टाइमर और +2 / −0.5 की वही मार्किंग जो असली परीक्षा में होती है।
याद रखने योग्य नियम: गुणा में घातांक जुड़ते हैं, भाग में घटते हैं, कोष्ठक में गुणा होते हैं; किसी भी शून्येतर संख्या की घात शून्य 1 होती है, पर 0k = 0; ऋणात्मक घातांक भिन्न को पलट देता है। करणी में √(a + 2√b) = √m + √n जहाँ m + n = a और mn = b; और अनंत मूल √(a + √(a + …)) का मान n + 1 होता है जब a = n(n+1) हो।
पूरे नियम हिंदी में पढ़ने के लिए घात, घातांक एवं करणी के अध्ययन नोट्स देखिए।