1. Nature of Mixed Square & Square Root Problems
Competitive examinations often combine squares, square roots, fractions, decimals, identities and arithmetic operations in a single question. Such questions are usually easier when the expression is simplified in the correct order instead of calculating every value separately.
Exam Strategy:
Before calculating, check whether the expression contains:
• A known perfect square
• A useful algebraic identity
• A decimal or fraction that can be converted easily
• Consecutive squares
• An approximation opportunity
• Options that can be eliminated quickly
2. Square and Square Root as Inverse Operations
Key Relationship:
For a ≥ 0:
(√a)2 = a
For any real a:
√(a2) = |a|
Examples:
(√81)2 = 81
√(152) = 15
√((-15)2) = 15
Exam Trap:
For a general real number a, √(a2) is |a|, not always a.
3. BODMAS with Squares and Square Roots
Order of Operations:
Brackets → Orders/Roots → Division and Multiplication → Addition and Subtraction
Example:
√144 + 52 ÷ 5
= 12 + 25 ÷ 5
= 12 + 5
= 17
Exam Tip:
Squares and square roots should normally be evaluated before ordinary addition and subtraction.
4. Equations Involving a Square
Rule:
If:
x2 = N,
then over the real numbers:
x = ±√N
Example:
x2 = 2304
x = ±√2304
= ±48
But:
If the question states x > 0, then x = 48 only.
5. Difference of Two Squares under a Square Root
Useful Identity:
a2 − b2 = (a+b)(a−b)
Example:
√(252−72)
= √[(25+7)(25−7)]
= √(32×18)
= √576
= 24
Shortcut:
Using the identity is often faster than calculating both squares separately.
6. Mixed Sum and Difference of Square Roots
Method:
Evaluate exact perfect-square roots first, then perform addition or subtraction.
Example:
√256 + √81 − √49
= 16 + 9 − 7
= 18
Common Mistake:
Do not combine the radicands directly.
√256 + √81 is not √337.
7. Mixed Decimal Square-Root Expressions
Example:
√0.81 + √2.25
= 0.9 + 1.5
= 2.4
Shortcut:
Convert decimal perfect squares into familiar forms mentally wherever possible.
Example:
√0.0081 + √0.81
= 0.09 + 0.9
= 0.99
8. Mixed Fraction Square-Root Expressions
Rule:
For non-negative numerator and positive denominator:
√(a/b) = √a/√b
Example:
√(144/225) + √(49/100)
= 12/15 + 7/10
= 4/5 + 7/10
= 8/10 + 7/10
= 3/2
Exam Strategy:
Simplify individual fractions before combining them.
9. Identifying the Greatest Perfect Square Below a Number
Method:
Find the greatest integer n such that:
n2 ≤ N.
Then n2 is the greatest perfect square not exceeding N.
Example:
Find the greatest perfect square less than or equal to 999.
312 = 961
322 = 1024
Therefore the required number is 961.
10. Greatest n-Digit Perfect Square
Rule:
The greatest n-digit number is:
10n−1.
Take its square root, discard the fractional part and square the resulting integer.
Example:
Greatest 4-digit number = 9999.
√9999 lies between 99 and 100.
Therefore greatest 4-digit perfect square:
992 = 9801.
11. Smallest n-Digit Perfect Square
Method:
The smallest n-digit number is 10n−1.
Find the smallest integer whose square is at least this number.
Example:
Find the smallest 4-digit perfect square.
Smallest 4-digit number = 1000.
312 = 961
322 = 1024
Therefore the smallest 4-digit perfect square is 1024.
12. Number of Digits in the Square Root of a Perfect Square
Rule:
If a positive perfect square has d digits, then its positive integer square root has:
ceil(d/2) digits.
Equivalent Rule:
A perfect square having either:
2n−1 digits or 2n digits
has an n-digit positive square root.
Examples:
729 has 3 digits → √729 = 27 has 2 digits.
9604 has 4 digits → √9604 = 98 has 2 digits.
74529 has 5 digits → √74529 = 273 has 3 digits.
13. Moving from One Perfect Square to Another
Rules:
(n+1)2 = n2 + 2n + 1
(n−1)2 = n2 − (2n−1)
Example:
If 452 = 2025, then:
462
= 2025 + 91
= 2116
Shortcut:
This is much faster than multiplying 46×46 from the beginning.
14. Comparing Squares without Full Multiplication
Property:
For non-negative numbers a and b:
a > b ⇔ a2 > b2
Example:
Without calculating completely, compare 512 and 492.
Since 51 > 49:
512 > 492.
Important:
When negative numbers are involved, compare absolute values before comparing their squares.
15. Comparing Square Roots without Finding Exact Values
Property:
For non-negative a and b:
a > b ⇔ √a > √b
Example:
√85 > √80 because 85 > 80.
Exam Tip:
If only comparison is required, there is usually no need to calculate decimal approximations.
16. Approximation in Mixed Square-Root Expressions
Method:
When an approximate answer is required, replace non-perfect square roots by nearby convenient values before performing the remaining operations.
Example:
Approximate:
√1025 + 28 − √170
√1025 ≈ 32
√170 ≈ 13
Therefore:
32 + 28 − 13
= 47
Exam Tip:
Approximation should be used only when the question or answer choices permit approximation.
17. Checking Whether an Answer Is Reasonable
Quick Checks:
• Compare the answer with nearby squares.
• Check the unit digit when an exact integer square root is expected.
• Square the proposed root when verification is easy.
• Estimate the size before detailed calculation.
• Check decimal placement carefully.
• If the answer is to x2=N, check whether both positive and negative roots are required.
18. Common Exam Traps
Trap 1:
Writing x=√N instead of x=±√N when solving x2=N without any sign restriction.
Trap 2:
Using √a+√b=√(a+b).
Trap 3:
Ignoring BODMAS in expressions containing roots and powers.
Trap 4:
Choosing 9999 itself as the greatest 4-digit perfect square.
Trap 5:
Using the previous square when the smallest higher perfect square is required.
Trap 6:
Calculating exact square roots in a question that only requires approximate comparison.
Trap 7:
Misplacing the decimal point in decimal square roots.
Trap 8:
Forgetting that √(a2)=|a|.
19. Quick Revision
Remember:
• Squares and square roots are inverse operations for non-negative quantities.
• √(a2)=|a|.
• In mixed expressions, follow BODMAS.
• If x2=N, then x=±√N unless a sign condition is given.
• Use a2−b2=(a+b)(a−b) when helpful.
• Evaluate exact perfect-square roots before combining them.
• Greatest perfect square ≤N is obtained from floor(√N)2.
• Smallest perfect square ≥N is obtained from ceil(√N)2 when N itself is not already a perfect square.
• Greatest 4-digit perfect square is 9801.
• Smallest 4-digit perfect square is 1024.
• A perfect square with 2n−1 or 2n digits has an n-digit positive integer square root.
• Consecutive squares differ by consecutive odd numbers.
• Exact calculation is unnecessary when only comparison or approximation is asked.
• Always estimate the expected size of the answer.
20. Verified Previous-Year Questions
RRB Group D PYQ
Difference of Squares
2 September 2022 · Shift 1
Q1. What is the positive square root of (342−162)?
A. 50
B. 40
C. 30
D. 60
Correct Answer: C. 30
342−162
=(34+16)(34−16)
=50×18
=900.
Therefore:
√900=30.
RRB Group D PYQ
Approximation
6 September 2022 · Shift 2
Q2. Find the closest approximate value of:
√1025 + 98 ÷ 14 × 4 − √170
A. 70
B. 47
C. 27
D. 60
Correct Answer: B. 47
√1025 ≈ 32
√170 ≈ 13
98÷14×4
=7×4
=28.
Therefore:
32+28−13
=47.
RRB NTPC PYQ
Mixed Decimal Roots
8 August 2025 · UG CBT-I · Shift 1
Q3. Evaluate:
√256 + √0.0256 − √4.41
A. 14.06
B. 13.37
C. 13.18
D. 7.4
Correct Answer: A. 14.06
√256=16
√0.0256=0.16
√4.41=2.1
Therefore:
16+0.16−2.1
=14.06.
MP Patwari PYQ
Greatest Perfect Square
9 December 2017 · Shift 2
Q4. Which is the greatest 4-digit number that is a perfect square?
A. 9999
B. 9909
C. 9801
D. 9081
Correct Answer: C. 9801
The greatest 4-digit number is 9999.
992=9801
1002=10000.
Therefore the greatest 4-digit perfect square is 9801.
21. Practice MCQs
Practice MCQ
Q1. Evaluate √(252−72).
A. 18
B. 20
C. 24
D. 32
Correct Answer: C. 24
(25²−7²)=(25+7)(25−7)=32×18=576. Therefore √576=24.
Practice MCQ
Q2. Find √0.81 + √2.25.
A. 1.6
B. 2.1
C. 2.4
D. 2.6
Correct Answer: C. 2.4
√0.81=0.9 and √2.25=1.5. Therefore total=2.4.
Practice MCQ
Q3. If x>0 and x2=2304, find x.
A. 46
B. 47
C. 48
D. 49
Correct Answer: C. 48
√2304=48. Since x>0, x=48.
Practice MCQ
Q4. What is the greatest 3-digit perfect square?
A. 900
B. 961
C. 980
D. 999
Correct Answer: B. 961
31²=961 and 32²=1024. Therefore 961 is the greatest 3-digit perfect square.
Practice MCQ
Q5. What is the smallest 4-digit perfect square?
A. 1000
B. 1001
C. 1024
D. 1089
Correct Answer: C. 1024
31²=961 and 32²=1024. Hence 1024 is the smallest 4-digit perfect square.
Practice MCQ
Q6. A positive perfect square has 7 digits. How many digits can its positive integer square root have?
A. 2
B. 3
C. 4
D. 5
Correct Answer: C. 4
A perfect square with 2n−1=7 digits has n=4 digits in its positive integer square root.
Practice MCQ
Q7. Evaluate √(144/225)+√(49/100).
A. 13/10
B. 3/2
C. 17/10
D. 19/10
Correct Answer: B. 3/2
√(144/225)=4/5 and √(49/100)=7/10. Therefore 4/5+7/10=8/10+7/10=3/2.
Practice MCQ
Q8. √410 is closest to which integer?
A. 18
B. 19
C. 20
D. 21
Correct Answer: C. 20
20²=400 and 21²=441. Since 410 is much closer to 400, √410 is closest to 20.
Practice MCQ
Q9. If 452=2025, what is 462?
A. 2070
B. 2091
C. 2116
D. 2136
Correct Answer: C. 2116
46²=45²+(2×45+1)=2025+91=2116.
Practice MCQ
Q10. Find √0.0081+√0.81.
A. 0.81
B. 0.90
C. 0.99
D. 1.09
Correct Answer: C. 0.99
√0.0081=0.09 and √0.81=0.9. Their sum=0.99.
1. Mixed Square एवं Square Root Problems की प्रकृति
Competitive examinations में squares, square roots, fractions, decimals, identities तथा arithmetic operations को एक ही question में combine किया जाता है। ऐसे questions में lengthy calculation के बजाय सही method चुनना अधिक महत्वपूर्ण होता है।
Exam Strategy:
Calculation शुरू करने से पहले देखें कि expression में:
• कोई known perfect square है या नहीं
• कोई useful identity लग सकती है या नहीं
• Decimal या fraction को आसानी से convert किया जा सकता है या नहीं
• Consecutive squares का उपयोग हो सकता है या नहीं
• Approximation पर्याप्त है या नहीं
• Options से elimination संभव है या नहीं
2. Square एवं Square Root का Inverse Relation
Key Relationship:
a ≥ 0 के लिए:
(√a)2 = a
किसी भी real a के लिए:
√(a2) = |a|
उदाहरण:
(√81)2 = 81
√(152) = 15
√((-15)2) = 15
Exam Trap:
General real number a के लिए √(a2) का सही result |a| है।
3. Squares एवं Square Roots में BODMAS
Order:
Brackets → Orders/Roots → Division एवं Multiplication → Addition एवं Subtraction
उदाहरण:
√144 + 52 ÷ 5
= 12 + 25 ÷ 5
= 12 + 5
= 17
Exam Tip:
Addition या subtraction से पहले powers एवं roots evaluate करें।
4. Square वाली Equations
Rule:
यदि:
x2 = N,
तो real numbers में:
x = ±√N
उदाहरण:
x2 = 2304
x = ±√2304
= ±48
लेकिन:
यदि x>0 दिया गया हो, तो केवल x=48 होगा।
5. Difference of Squares के अंदर Square Root
Useful Identity:
a2−b2=(a+b)(a−b)
उदाहरण:
√(252−72)
=√[(25+7)(25−7)]
=√(32×18)
=√576
=24
Shortcut:
दोनों squares अलग-अलग calculate करने के बजाय difference-of-squares identity लगाना अक्सर faster होता है।
6. Square Roots का Mixed Addition एवं Subtraction
उदाहरण:
√256 + √81 − √49
=16+9−7
=18
Common Mistake:
√256+√81 को √337 न लिखें।
7. Decimal Square Roots के Mixed Expressions
उदाहरण:
√0.81+√2.25
=0.9+1.5
=2.4
उदाहरण:
√0.0081+√0.81
=0.09+0.9
=0.99
Shortcut:
Familiar decimal perfect squares को mentally identify करें।
8. Fraction Square Roots के Mixed Expressions
Rule:
a ≥ 0 एवं b>0 के लिए:
√(a/b)=√a/√b
उदाहरण:
√(144/225)+√(49/100)
=4/5+7/10
=8/10+7/10
=3/2
9. किसी Number से छोटा Greatest Perfect Square
Method:
ऐसा greatest integer n खोजें जिसके लिए:
n2 ≤ N।
उदाहरण:
999 से कम या बराबर greatest perfect square:
312=961
322=1024
इसलिए answer=961।
10. Greatest n-Digit Perfect Square
Rule:
Greatest n-digit number = 10n−1।
उसका square root estimate करें, fractional part हटाएँ और प्राप्त integer का square करें।
उदाहरण:
Greatest 4-digit number=9999।
992=9801
1002=10000
इसलिए greatest 4-digit perfect square=9801।
11. Smallest n-Digit Perfect Square
Method:
Smallest n-digit number=10n−1।
ऐसा smallest integer चुनें जिसका square इस number के बराबर या उससे बड़ा हो।
उदाहरण:
Smallest 4-digit number=1000।
312=961
322=1024
इसलिए smallest 4-digit perfect square=1024।
12. Square Root में Digits की संख्या
Rule:
यदि positive perfect square में d digits हैं, तो उसके positive integer square root में:
ceil(d/2) digits होंगी।
Equivalent Rule:
2n−1 या 2n digits वाले perfect square के positive square root में n digits होती हैं।
उदाहरण:
729 → 3 digits → √729=27 → 2 digits
9604 → 4 digits → √9604=98 → 2 digits
74529 → 5 digits → √74529=273 → 3 digits
13. एक Perfect Square से दूसरे Perfect Square तक
Rules:
(n+1)2=n2+2n+1
(n−1)2=n2−(2n−1)
उदाहरण:
यदि 452=2025 है, तो:
462
=2025+91
=2116।
14. Full Multiplication के बिना Squares की Comparison
Property:
Non-negative a एवं b के लिए:
a>b ⇔ a2>b2
Important:
Negative numbers होने पर squares compare करने से पहले absolute values पर ध्यान दें।
15. Exact Value के बिना Square Roots की Comparison
Property:
Non-negative a एवं b के लिए:
a>b ⇔ √a>√b
उदाहरण:
85>80 होने के कारण:
√85>√80।
Exam Tip:
यदि केवल comparison पूछा गया है, तो decimal square roots निकालना unnecessary है।
16. Mixed Square-Root Expressions में Approximation
उदाहरण:
Approximate करें:
√1025 + 28 − √170
√1025≈32
√170≈13
इसलिए:
32+28−13=47।
Exam Tip:
Approximation केवल तभी use करें जब question या answer options approximate answer allow करते हों।
17. Answer की Quick Verification
Quick Checks:
• Answer को nearby squares से compare करें।
• Integer square root में unit digit check करें।
• आवश्यकता होने पर proposed root का square करें।
• Detailed calculation से पहले expected magnitude estimate करें।
• Decimal position check करें।
• x2=N में देखें कि ± दोनों roots चाहिए या केवल positive root।
18. Common Exam Traps
Trap 1:
x2=N में बिना sign restriction के x=√N लिखना; सामान्यतः x=±√N होगा।
Trap 2:
√a+√b=√(a+b) मान लेना।
Trap 3:
Roots एवं powers वाले expressions में BODMAS ignore करना।
Trap 4:
9999 को greatest 4-digit perfect square मान लेना।
Trap 5:
Smallest higher perfect square पूछे जाने पर previous square चुन लेना।
Trap 6:
Approximation वाले question में unnecessary exact root निकालना।
Trap 7:
Decimal square root में decimal point गलत रखना।
Trap 8:
√(a2)=|a| को भूल जाना।
19. Quick Revision
एक नज़र में:
• Squares एवं square roots non-negative quantities के लिए inverse operations हैं।
• √(a2)=|a|।
• Mixed expressions में BODMAS follow करें।
• x2=N में सामान्यतः x=±√N।
• Difference of squares के लिए a2−b2=(a+b)(a−b) use करें।
• Exact perfect-square roots पहले evaluate करें।
• Greatest perfect square ≤N के लिए floor(√N)2 use किया जा सकता है।
• Smallest required higher perfect square के लिए next integer square देखें।
• Greatest 4-digit perfect square=9801।
• Smallest 4-digit perfect square=1024।
• 2n−1 या 2n digits वाले perfect square का positive integer root n-digit होता है।
• Consecutive squares में difference odd numbers का होता है।
• केवल comparison/approximation पूछे जाने पर exact calculation unnecessary हो सकती है।
• Final answer के expected size को जरूर check करें।
20. Verified Previous-Year Questions
RRB Group D PYQ
Difference of Squares
2 सितंबर 2022 · Shift 1
प्रश्न 1. (342−162) का positive square root क्या है?
A. 50
B. 40
C. 30
D. 60
सही उत्तर: C. 30
342−162
=(34+16)(34−16)
=50×18
=900।
इसलिए √900=30।
RRB Group D PYQ
Approximation
6 सितंबर 2022 · Shift 2
प्रश्न 2. निम्न expression का closest approximate value ज्ञात करें:
√1025 + 98 ÷ 14 × 4 − √170
A. 70
B. 47
C. 27
D. 60
सही उत्तर: B. 47
√1025≈32 तथा √170≈13।
98÷14×4=7×4=28।
इसलिए:
32+28−13=47।
RRB NTPC PYQ
Mixed Decimal Roots
8 अगस्त 2025 · UG CBT-I · Shift 1
प्रश्न 3. निम्न का value ज्ञात करें:
√256 + √0.0256 − √4.41
A. 14.06
B. 13.37
C. 13.18
D. 7.4
सही उत्तर: A. 14.06
√256=16
√0.0256=0.16
√4.41=2.1
इसलिए:
16+0.16−2.1=14.06।
MP Patwari PYQ
Greatest Perfect Square
9 दिसंबर 2017 · Shift 2
प्रश्न 4. Greatest 4-digit perfect-square number कौन-सा है?
A. 9999
B. 9909
C. 9801
D. 9081
सही उत्तर: C. 9801
Greatest 4-digit number=9999।
992=9801
1002=10000।
इसलिए greatest 4-digit perfect square=9801।
21. Practice MCQs
Practice MCQ
प्रश्न 1. √(252−72) ज्ञात करें।
A. 18
B. 20
C. 24
D. 32
सही उत्तर: C. 24
(25²−7²)=(25+7)(25−7)=576। इसलिए √576=24। Practice MCQ
प्रश्न 2. √0.81+√2.25 ज्ञात करें।
A. 1.6
B. 2.1
C. 2.4
D. 2.6
सही उत्तर: C. 2.4
√0.81=0.9 एवं √2.25=1.5। Sum=2.4। Practice MCQ
प्रश्न 3. यदि x>0 और x2=2304 है, तो x ज्ञात करें।
A. 46
B. 47
C. 48
D. 49
सही उत्तर: C. 48
√2304=48 और x positive है। इसलिए x=48। Practice MCQ
प्रश्न 4. Greatest 3-digit perfect square कौन-सा है?
A. 900
B. 961
C. 980
D. 999
सही उत्तर: B. 961
31²=961 जबकि 32²=1024। इसलिए answer=961। Practice MCQ
प्रश्न 5. Smallest 4-digit perfect square कौन-सा है?
A. 1000
B. 1001
C. 1024
D. 1089
सही उत्तर: C. 1024
31²=961 एवं 32²=1024। Practice MCQ
प्रश्न 6. किसी positive perfect square में 7 digits हैं। उसके positive integer square root में कितनी digits होंगी?
A. 2
B. 3
C. 4
D. 5
सही उत्तर: C. 4
7=2n−1 रखने पर n=4। इसलिए square root में 4 digits होंगी। Practice MCQ
प्रश्न 7. √(144/225)+√(49/100) ज्ञात करें।
A. 13/10
B. 3/2
C. 17/10
D. 19/10
सही उत्तर: B. 3/2
√(144/225)=4/5 और √(49/100)=7/10। Sum=8/10+7/10=3/2। Practice MCQ
प्रश्न 8. √410 किस integer के सबसे निकट है?
A. 18
B. 19
C. 20
D. 21
सही उत्तर: C. 20
20²=400 तथा 21²=441। 410, 400 के अधिक निकट है। इसलिए √410 का nearest integer 20 है। Practice MCQ
प्रश्न 9. यदि 452=2025 है, तो 462 कितना होगा?
A. 2070
B. 2091
C. 2116
D. 2136
सही उत्तर: C. 2116
46²=45²+(2×45+1)=2025+91=2116। Practice MCQ
प्रश्न 10. √0.0081+√0.81 ज्ञात करें।
A. 0.81
B. 0.90
C. 0.99
D. 1.09
सही उत्तर: C. 0.99
√0.0081=0.09 एवं √0.81=0.9। Sum=0.99।