Quantitative Aptitudeमात्रात्मक योग्यता · करणी और घातांक
Surds and Indicesकरणी और घातांकquestions and answersप्रश्न और उत्तर
Indices and surds are the grammar of quick calculation. Once the handful of rules below are automatic, expressions that look heavy, such as (8/27)^(−2/3), take a few seconds: the cube root is 2/3, squared is 4/9, and the minus sign flips it to 9/4.
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SSC CGL ~2
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Practice questions are in English for now.अभ्यास के प्रश्न अभी अंग्रेज़ी में हैं।
How surds and indices worksकरणी और घातांक को समझिए
The laws of indices work only when the base is the same. Then multiplication adds the powers, division subtracts them, and a power of a power multiplies them. To solve an equation such as 9^(x + 1) = 3⁸, first rewrite 9 as 3², so 2(x + 1) = 8 and x = 3.
A negative power is a reciprocal, never a negative number: 2⁻³ = 1/8. A fractional power p/q means “take the q-th root, then raise to the power p”, and taking the root first keeps the numbers small: 32^(3/5) = 2³ = 8.
A surd is a root that cannot be simplified to a whole number, such as √2 or √7. Surds combine only when the number under the root is the same, so simplify each one first: √50 + √18 = 5√2 + 3√2 = 8√2. To clear a surd from a denominator, multiply top and bottom by the conjugate, since (√a + √b)(√a − √b) = a − b.
Each trick says when to use it, why it works and when it fails. Formula tricks are checked by code over their whole range.हर ट्रिक बताती है कि उसे कब इस्तेमाल करें, वह क्यों काम करती है और कब नहीं। सूत्र वाली ट्रिक्स को कोड से उनकी पूरी रेंज पर जाँचा गया है।
Mental mathsमानसिक गणित✓ Checked on 7,569 cases✓ 7,569 मामलों पर जाँची गई
Compare roots of different orders by raising to the LCM power
Use it when:कब इस्तेमाल करें: “Which is the largest: √3, ∛4, ⁴√6?” or “arrange in ascending order” with roots of different orders.
Take the LCM of the root orders (2, 3, 4 → 12).
Raise each surd to that power: (√3)¹² = 3⁶ = 729, (∛4)¹² = 4⁴ = 256, (⁴√6)¹² = 6³ = 216.
Order the whole numbers; the surds are in the same order.
Exampleउदाहरण
Which is larger, √2 or ∛3?
→ LCM 6: (√2)⁶ = 8, (∛3)⁶ = 9. So ∛3 is larger.
Why it works, and when it failsयह क्यों काम करती है, और कब नहीं
Why:क्यों: For positive numbers, x < y exactly when xᴸ < yᴸ for any positive power L, because raising to a positive power keeps the order. Choosing L as the LCM turns every root into a whole number.
Limits:सीमाएँ: Positive numbers only. With many surds the powers grow quickly; compare two at a time if the numbers get large.
Solved examplesहल किए गए उदाहरण
Answers and full solutions are shown. Hide them to test yourself first.उत्तर और पूरे हल दिखाए गए हैं। पहले खुद को परखने के लिए इन्हें छिपाइए।
Example 1उदाहरण 1 · Laws of indices and surds · Easyआसान
If (6²)⁶ ÷ 6⁶ = 6ˣ, find x.
2
6
18
58
Answer:उत्तर:6
(6²)⁶ = 6¹².
6¹² ÷ 6⁶ = 6⁶.
So x = 6.
Example 2उदाहरण 2 · Laws of indices and surds · Mediumमध्यम
If 36^(x − 3) = 6¹⁰, find x.
5
13/2
8
13
Answer:उत्तर:8
36^(x − 3) = 6^(2x − 6).
2x − 6 = 10 ⇒ 2x = 16.
x = 8.
Example 3उदाहरण 3 · Laws of indices and surds · Hardकठिन
Rationalise the denominator and simplify: 11/(√13 − √2).