Quantitative Aptitudeमात्रात्मक योग्यता · वर्गमूल और घनमूल
Square and Cube Rootsवर्गमूल और घनमूलquestions and answersप्रश्न और उत्तर
Exam papers ask for roots of perfect squares and cubes far more often than for approximations, and for those there is no need for long division. The last digit narrows the answer to one or two candidates, and a quick bracket of the leading digits fixes the rest.
Unlimitedअसीमितfresh questionsनए प्रश्न
Answersउत्तरalways one tap awayहमेशा एक टैप दूर
1subtopicउप-टॉपिक
SSC CGL ~2
IBPS PO ~1
Practice questions are in English for now.अभ्यास के प्रश्न अभी अंग्रेज़ी में हैं।
How square and cube roots worksवर्गमूल और घनमूल को समझिए
For a square root, look at the last digit first: a square ending in 6 has a root ending in 4 or 6. Then drop the last two digits and find the largest square not bigger than what is left; for 7,056 that is 70, which lies between 8² and 9², so the root is in the 80s. Since 85² = 7,225 is more than 7,056, the root is 84.
For a cube root of up to six digits, the last digit decides the last digit of the root uniquely, because the cubes of 0 to 9 all end in different digits. Strike off the last three digits and bracket what is left between two cubes: for 1,40,608 the last digit 8 gives 2, and 140 lies between 5³ = 125 and 6³ = 216, so the root is 52.
Decimals follow the same method once you count places. Pair the digits from the decimal point: a number with four decimal places has a square root with two. So √0.0729 = 0.27 because √729 = 27, and √12.25 = 3.5 because √1225 = 35.
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 999 cases✓ 999 मामलों पर जाँची गई
Square root of a perfect square: unit digit, bracket, then the “5” test
Use it when:कब इस्तेमाल करें: The square root of a perfect square of up to six digits is asked, or an option must be checked as a square.
Last digit → possible last digits of the root: 1 → 1/9, 4 → 2/8, 9 → 3/7, 6 → 4/6, 5 → 5, 0 → 0.
Strike off the last two digits; find the largest square not more than what is left. Its root gives the leading digit(s).
If two candidates remain, compare the number with the square of the middle one ending in 5: below it → the smaller candidate, above it → the larger.
Exampleउदाहरण
√7,056
→ Ends in 6 → 4 or 6. 70 lies between 8² and 9² → 84 or 86. 85² = 7,225 > 7,056 → 84.
Why it works, and when it failsयह क्यों काम करती है, और कब नहीं
Why:क्यों: A square’s last digit depends only on the root’s last digit, and (10t)² ≤ N < (10t + 10)² pins down the leading part t. Between 10t + d and 10t + 10 − d, the square of 10t + 5 lies in the middle, so the comparison picks the right one.
Limits:सीमाएँ: Only for exact perfect squares. A non-square ending in 1, 4, 5, 6, 9 or 0 still gives a confident but wrong answer, so check by squaring when in doubt.
Mental mathsमानसिक गणित✓ Checked on 999 cases✓ 999 मामलों पर जाँची गई
Cube root of a perfect cube from its last digit and its thousands
Use it when:कब इस्तेमाल करें: The cube root of a perfect cube of up to six digits, such as ∛3,14,432 or ∛39,304.
Last digit of the cube → last digit of the root. 2 and 8 swap, 3 and 7 swap, every other digit stays.
Strike off the last three digits; find the largest cube not more than what is left. Its root is the tens digit.
Exampleउदाहरण
∛3,14,432
→ Ends in 2 → root ends in 8. 314 lies between 6³ = 216 and 7³ = 343 → 6. Root = 68.
Why it works, and when it failsयह क्यों काम करती है, और कब नहीं
Why:क्यों: The cubes of 0 to 9 end in 0, 1, 8, 7, 4, 5, 6, 3, 2, 9: all different, so the last digit of a cube fixes the last digit of its root. Since (10t)³ = 1000t³, striking off three digits and bracketing gives t.
Limits:सीमाएँ: Perfect cubes only, and the two-step version handles roots up to 99 (cubes up to six digits). Larger roots need the bracket step done on a longer leading part.
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 · Square and cube roots by the unit-digit method · Easyआसान
Lakshmi arranges 3,969 chairs in a perfect square for a school function. How many chairs are in each row?
53
63
67
73
Answer:उत्तर:63
Unit digit 9 → root ends in 3 or 7.
39 lies between 6² and 7², so the root is 60-something.
Choose between 63 and 67: 65² = 4,225 > 3,969, so the root is 63.
Check: 63² = 3,969.
Example 2उदाहरण 2 · Square and cube roots by the unit-digit method · Mediumमध्यम
A cubical water tank in Ranchi holds 3,89,017 cubic decimetres. What is the length of its edge in decimetres?
63
73
77
83
Answer:उत्तर:73
Last digit 7 → root ends in 3.
389 lies between 7³ and 8³ → tens digit 7.
Root = 73. Check: 73³ = 3,89,017.
Example 3उदाहरण 3 · Square and cube roots by the unit-digit method · Hardकठिन
Find √0.7225.
0.085
0.3613
0.85
8.5
Answer:उत्तर:0.85
√7,225 = 85 (unit digit and bracketing).
0.7225 = 7,225/10,000, so √0.7225 = 85/100 = 0.85.
Example 4उदाहरण 4 · Square and cube roots by the unit-digit method · Mediumमध्यम
Arjun arranges 13,225 chairs in a perfect square for a school function. How many chairs are in each row?
105
115
117
125
Answer:उत्तर:115
Unit digit 5 → root ends in 5.
132 lies between 11² and 12², so the root is 110-something.
Only one candidate fits: 115.
Check: 115² = 13,225.
Example 5उदाहरण 5 · Square and cube roots by the unit-digit method · Mediumमध्यम
Aditya arranges 41,209 chairs in a perfect square for a school function. How many chairs are in each row?
193
203
207
213
Answer:उत्तर:203
Unit digit 9 → root ends in 3 or 7.
412 lies between 20² and 21², so the root is 200-something.
Choose between 203 and 207: 205² = 42,025 > 41,209, so the root is 203.
Check: 203² = 41,209.
Example 6उदाहरण 6 · Square and cube roots by the unit-digit method · Mediumमध्यम
Find ∛54,872.
28
32
38
48
Answer:उत्तर:38
Last digit 2 → root ends in 8.
54 lies between 3³ and 4³ → tens digit 3.
Root = 38. Check: 38³ = 54,872.
Common mistakesआम गलतियाँ
Picking the wrong member of a unit-digit pair (84 or 86) without comparing with the square of the middle number ending in 5.
Using the square-root pairs (2 or 8, 3 or 7) for cube roots, where 2 and 8, and 3 and 7, simply swap.
Putting the decimal point in the wrong place by not halving the number of decimal places.
Frequently asked questionsअक्सर पूछे जाने वाले प्रश्न
Can a perfect square end in 2, 3, 7 or 8?
No. The squares of 0 to 9 end only in 0, 1, 4, 5, 6 or 9, so an option ending in 2, 3, 7 or 8 cannot be a perfect square.
How do I find the cube root of a six-digit perfect cube quickly?
Take the last digit of the root from the last digit of the cube (swapping 2 ↔ 8 and 3 ↔ 7), then strike off the last three digits and find the largest cube not bigger than the rest. That cube’s root is the first digit.
Does this method work for numbers that are not perfect squares?
No. It only identifies exact roots. For other numbers, bracket between two squares and estimate, for example √50 is a little more than 7 because 7² = 49.