Topic 2.4 : Simplification with Powers, Squares, Cubes & Roots Topic 2.4 : घात, वर्ग, घन एवं मूल वाले व्यंजकों का सरलीकरण

Learn how to simplify mathematical expressions involving powers, exponents, squares, cubes, square roots and cube roots for JSSC, JPSC, SSC, Railway and other competitive examinations. Understand exponent laws, powers of powers, zero and negative exponents, perfect squares and cubes, simplification of roots, common square and cube values, BODMAS applications, calculation shortcuts, common exam traps, verified PYQs and practice MCQs. JSSC, JPSC, SSC, Railway एवं अन्य प्रतियोगी परीक्षाओं के लिए powers, exponents, squares, cubes, square roots और cube roots वाले mathematical expressions का simplification सीखें। Exponent laws, power of a power, zero एवं negative exponents, perfect squares और cubes, roots का simplification, महत्वपूर्ण square-cube values, BODMAS applications, calculation shortcuts, common exam traps, verified PYQs और practice MCQs इस topic में शामिल हैं।

Chapter 2 : Simplification & Approximation अध्याय 2 : सरलीकरण एवं सन्निकटन

1. Powers and Roots in Simplification

Powers, squares, cubes and roots frequently appear inside competitive-exam simplification expressions. According to the order of operations, powers and roots are evaluated before ordinary multiplication, division, addition and subtraction unless a bracket changes the grouping.

This topic focuses on their use in simplification. Detailed theory of squares, cubes, surds and indices is covered separately in later chapters.

Key Rule:
In BODMAS, powers and roots belong to the Orders stage. They are evaluated after brackets but before ordinary multiplication, division, addition and subtraction.
Example:
Simplify: 12 + 32 × 4

32 = 9
9×4 = 36
12+36 = 48.
Common Mistake:
Do not perform addition before evaluating the power. In 12+32×4, the value 32 must be found first.

2. Understanding Exponent Notation

In an, the number a is called the base and n is the exponent or power.

Basic Meaning:
an = a×a×a×...×a, where a is multiplied n times.
Example:
25 = 2×2×2×2×2 = 32.

53 = 5×5×5 = 125.
Observation:
a2 is called the square of a, while a3 is called the cube of a.

3. Sign and Power: A Very Important Distinction

Brackets are extremely important when negative numbers are raised to powers.

Key Difference:
(−3)2 = (−3)(−3) = 9

but

−32 = −(32) = −9.
Example:
Simplify:
(−2)4 − 24

(−2)4 = 16
24 = 16

Therefore:
16−16 = 0.
Exam Trap:
Never assume that −a2 and (−a)2 are the same. The position of the brackets changes the meaning.

4. Multiplication of Powers with the Same Base

When powers having the same base are multiplied, their exponents are added.

Exponent Rule:
am × an = am+n
Example:
23 × 25

= 23+5
= 28
= 256.
Shortcut:
If the bases are the same, combine the exponents first instead of calculating the powers separately.

5. Division of Powers with the Same Base

When powers with the same non-zero base are divided, subtract the exponent in the denominator from the exponent in the numerator.

Exponent Rule:
am ÷ an = am−n, for a ≠ 0.
Example:
37 ÷ 34

= 37−4
= 33
= 27.
Exam Tip:
Before performing long division, check whether the numerator and denominator can be written with the same base.

6. Power of a Power

When a power itself is raised to another power, multiply the exponents.

Exponent Rule:
(am)n = amn
Example:
(23)4

= 212
= 4096.
Common Mistake:
Do not add the powers in (am)n. The exponents are multiplied.

7. Power of a Product and Quotient

Rules:
(ab)n = anbn

(a/b)n = an/bn, for b ≠ 0.
Example:
(2×3)3
= 23×33
= 8×27
= 216.
Observation:
These identities can also be used in reverse. For example:
24×54 = (2×5)4 = 104 = 10,000.

8. Zero and Negative Exponents

Important Rules:
a0 = 1, for a ≠ 0.

a−n = 1/an, for a ≠ 0.
Example:
2−3
= 1/23
= 1/8.
Example:
53 × 5−2
= 53−2
= 5.
Exam Trap:
A negative exponent does not make the value negative. It indicates the reciprocal.

9. Square Roots of Perfect Squares

The principal square root of a positive perfect square is the positive number whose square equals the given number.

Basic Relation:
If a≥0, then √(a2) = a.
Example:
√144 = 12
because 122 = 144.
Example:
√625 = 25
because 252 = 625.
Important Observation:
For a general real number a, √(a2) = |a|. For example, √((−5)2) = 5, not −5.

10. Simplifying Square Roots Using Perfect-Square Factors

If a number under a square root contains a perfect-square factor, that factor can be taken outside the root.

Key Rule:
√(a2b) = a√b, for a≥0 and b≥0.
Example:
√72

= √(36×2)
= 6√2.
Example:
√200

= √(100×2)
= 10√2.
Shortcut:
Look for the largest perfect-square factor inside the root. This usually gives the simplest form in one step.
Scope Note:
Detailed operations with surds and rationalisation are covered later in Chapter 7. Here the focus is only on roots needed for simplification.

11. Addition and Subtraction of Like Roots

Roots can be added or subtracted directly only after they have been reduced to the same radical part.

Example:
√50 + √8

= 5√2 + 2√2
= 7√2.
Example:
√75 − √27

= 5√3 − 3√3
= 2√3.
Exam Trap:
In general, √a + √b ≠ √(a+b).

For example:
√9 + √16 = 3+4 = 7,
while √25 = 5.

12. Cube Roots and Perfect Cubes

The cube root of a number is the value which, when cubed, gives the original number.

Basic Relation:
∛(a3) = a.
Example:
∛216 = 6
because 63 = 216.
Example:
∛(−125) = −5
because (−5)3 = −125.
Observation:
Unlike real square roots, real cube roots are defined for negative numbers as well.

13. Simplifying Cube Roots Using Perfect-Cube Factors

Key Rule:
∛(a3b) = a∛b.
Example:
∛54

= ∛(27×2)
= 3∛2.
Example:
∛250

= ∛(125×2)
= 5∛2.
Shortcut:
Look for perfect-cube factors such as 8, 27, 64, 125, 216, 343, 512 and 1000.

14. Roots of Products and Fractions

Square-Root Rules:
For a,b≥0:
√(ab) = √a × √b

For a≥0 and b>0:
√(a/b) = √a/√b
Example:
√(144/9)

= √144/√9
= 12/3
= 4.
Example:
∛(64×125)

= ∛64 × ∛125
= 4×5
= 20.
Common Mistake:
Product rules for roots do not mean that roots can be distributed across addition or subtraction.

15. Square Roots of Decimal Perfect Squares

A terminating decimal can often be converted into a fraction whose numerator and denominator are perfect squares.

Example:
√1.44

= √(144/100)
= 12/10
= 1.2.
Example:
√0.0081

= √(81/10000)
= 9/100
= 0.09.
Shortcut:
For a perfect-square decimal having an even number of decimal places, temporarily remove the decimal, find the square root and then place half as many decimal places in the result.
Exam Tip:
Use this shortcut only when the underlying integer is a perfect square. Otherwise estimate or use another appropriate method.

16. Combining Powers and Roots with BODMAS

When powers, roots and ordinary operations appear together, first resolve brackets, then powers and roots, and then continue with multiplication/division and addition/subtraction.

Example:
Simplify:
23 + √81 − ∛27

23 = 8
√81 = 9
∛27 = 3

Therefore:
8+9−3 = 14.
Example:
Simplify:
(52−32) ÷ √16

52−32 = 25−9 = 16
√16 = 4

16÷4 = 4.
Exam Method:
Mark each power and root mentally before starting. Evaluate all such “Orders” before proceeding to ordinary arithmetic.

17. Important Squares and Cubes for Fast Calculation

nn²nn³
1010011
1112128
12144327
13169464
141965125
152256216
162567343
172898512
183249729
19361101000
20400121728
25625153375
30900208000
Exam Shortcut:
For simplification questions, quick recall of common squares and cubes is often faster than applying a formal root-finding method.

18. Common Exam Traps

Trap 1:
Confusing −a² with (−a)².
Trap 2:
Adding exponents in a power of a power instead of multiplying them.
Trap 3:
Treating a negative exponent as a negative value rather than a reciprocal.
Trap 4:
Assuming √(a+b)=√a+√b.
Trap 5:
Forgetting that √(a²)=|a| for a real number a.
Trap 6:
Ignoring BODMAS when powers and roots appear inside a longer expression.
Trap 7:
Calculating large powers separately when the same-base exponent laws can simplify the expression immediately.

19. Quick Revision

Remember:
• Powers and roots are evaluated at the Orders stage of BODMAS.
• am×an = am+n.
• am÷an = am−n for a≠0.
• (am)n = amn.
• a0=1 for a≠0.
• a−n=1/an.
• √(a²)=|a|.
• Extract perfect-square factors from square roots.
• Extract perfect-cube factors from cube roots.
• Only like radicals can be directly added or subtracted.
• √(a+b) is generally not √a+√b.
• Memorise common squares and cubes for faster calculation.

20. Verified Previous-Year Questions

SSC MTS PYQ Squares & Square Root 3 May 2023 · Shift II

Q1. Find the value of √[((132−82)×5)+4].

A. 23
B. 26
C. 24
D. 22
Correct Answer: A. 23
132−82
=169−64
=105.

Then:
105×5+4
=525+4
=529.

√529 = 23.
RRB Group D PYQ Square Root of Decimal 30 August 2022 · Shift III

Q2. Find the value of √1.0201.

A. 10.1
B. 101
C. 1.01
D. 1.02
Correct Answer: C. 1.01
1.0201 = 10201/10000.

Since:
1012 = 10201
and 1002 = 10000,

√1.0201
=101/100
=1.01.
RRB NTPC PYQ Cube Root 26 July 2021 · CBT-I · Shift I

Q3. Between which two consecutive integers does ∛150 lie?

A. 6 and 7
B. 4 and 5
C. 7 and 8
D. 5 and 6
Correct Answer: D. 5 and 6
53 = 125
63 = 216.

Since:
125 < 150 < 216,

5 < ∛150 < 6.

Therefore ∛150 lies between 5 and 6.
RRB Group D PYQ Exponent Laws 5 February 2026 · Shift III

Q4. If √[32 × 64−2 × (2/163) × 4m+3] = 2, find m.

A. 8
B. 7
C. 6
D. 4
Correct Answer: B. 7
Write every term with base 2:

32=25
64−2=2−12
2/163=21−12=2−11
4m+3=22m+6.

Inside the square root:
25−12−11+2m+6
=22m−12.

Taking the square root:
2m−6 = 21.

Therefore:
m−6=1
m=7.

21. Practice MCQs

Practice MCQ

Q1. Simplify: 23 + √81 − ∛27

A. 12
B. 14
C. 16
D. 18
Correct Answer: B. 14
23=8, √81=9 and ∛27=3. Therefore 8+9−3=14.
Practice MCQ

Q2. Simplify: (52−32) ÷ √16

A. 2
B. 4
C. 6
D. 8
Correct Answer: B. 4
25−9=16 and √16=4. Therefore 16÷4=4.
Practice MCQ

Q3. Simplify: √144 + ∛216

A. 16
B. 17
C. 18
D. 20
Correct Answer: C. 18
√144=12 and ∛216=6. Sum=18.
Practice MCQ

Q4. Simplify: 43 ÷ 24

A. 2
B. 4
C. 6
D. 8
Correct Answer: B. 4
43=(22)3=26. Thus 26÷24=22=4.
Practice MCQ

Q5. Simplify: √(225/9)

A. 3
B. 5
C. 15
D. 25
Correct Answer: B. 5
√(225/9)=√25=5.
Practice MCQ

Q6. Simplify: (34 × 32) ÷ 35

A. 1
B. 3
C. 9
D. 27
Correct Answer: B. 3
34+2−5=31=3.
Practice MCQ

Q7. Simplify: √50 − √8

A. 2√2
B. 3√2
C. 5√2
D. 7√2
Correct Answer: B. 3√2
√50=5√2 and √8=2√2. Therefore 5√2−2√2=3√2.
Practice MCQ

Q8. Simplify: ∛(64×125)

A. 10
B. 15
C. 20
D. 25
Correct Answer: C. 20
∛64=4 and ∛125=5. Therefore 4×5=20.
Practice MCQ

Q9. Simplify: (√49)2 + ∛(−27)

A. 42
B. 44
C. 46
D. 52
Correct Answer: C. 46
√49=7, so (√49)2=49. Also ∛(−27)=−3. Therefore 49−3=46.
Practice MCQ

Q10. Simplify: 163/4

A. 4
B. 6
C. 8
D. 16
Correct Answer: C. 8
163/4=(161/4)3=23=8.

1. Simplification में Powers एवं Roots

Competitive examinations के simplification questions में powers, squares, cubes और roots बहुत frequently आते हैं। Order of operations के अनुसार brackets के बाद powers एवं roots solve किए जाते हैं और उसके बाद ordinary multiplication, division, addition तथा subtraction किया जाता है।

इस topic में इन concepts का focus केवल simplification पर है। Squares, cubes, surds तथा indices की detailed theory आगे अलग chapters में पढ़ी जाएगी।

मुख्य नियम:
BODMAS में powers और roots Orders stage में आते हैं। Brackets के बाद इन्हें ordinary multiplication, division, addition तथा subtraction से पहले solve करें।
उदाहरण:
12 + 32 × 4

32 = 9
9×4 = 36
12+36 = 48।
सामान्य गलती:
Power solve करने से पहले addition न करें। 12+32×4 में सबसे पहले 32 निकाला जाएगा।

2. Exponent Notation को समझना

an में a base तथा n exponent या power कहलाता है।

Basic Meaning:
an = a×a×a×...×a, जहाँ a को n बार multiply किया जाता है।
उदाहरण:
25 = 2×2×2×2×2 = 32।

53 = 5×5×5 = 125।
Observation:
a2 को a का square और a3 को a का cube कहा जाता है।

3. Sign और Power में महत्वपूर्ण अंतर

मुख्य अंतर:
(−3)2 = 9

लेकिन

−32 = −(32) = −9।
उदाहरण:
(−2)4 − 24

(−2)4 =16
24 =16

अतः:
16−16=0।
Exam Trap:
−a2 तथा (−a)2 को समान न समझें। Bracket की position mathematical meaning बदल देती है।

4. Same Base वाले Powers का Multiplication

Exponent Rule:
am × an = am+n
उदाहरण:
23 × 25

=28
=256।
Shortcut:
Bases same हों तो powers को अलग-अलग calculate करने के बजाय exponents पहले combine करें।

5. Same Base वाले Powers का Division

Exponent Rule:
am ÷ an = am−n, जहाँ a ≠ 0।
उदाहरण:
37 ÷ 34
=33
=27।
Exam Tip:
Long division करने से पहले देखें कि numerator और denominator को same base में लिखा जा सकता है या नहीं।

6. Power of a Power

Exponent Rule:
(am)n = amn
उदाहरण:
(23)4
=212
=4096।
सामान्य गलती:
Power of a power में exponents add नहीं होते; उन्हें multiply किया जाता है।

7. Product और Quotient की Power

Rules:
(ab)n = anbn

(a/b)n = an/bn, जहाँ b ≠ 0।
उदाहरण:
(2×3)3
=23×33
=8×27
=216।
Observation:
Rule को reverse भी कर सकते हैं:
24×54=(10)4=10,000।

8. Zero एवं Negative Exponents

महत्वपूर्ण Rules:
a0 = 1, जहाँ a ≠ 0।

a−n = 1/an, जहाँ a ≠ 0।
उदाहरण:
2−3
=1/23
=1/8।
उदाहरण:
53×5−2
=51
=5।
Exam Trap:
Negative exponent का अर्थ negative answer नहीं है। यह reciprocal दर्शाता है।

9. Perfect Squares के Square Roots

Basic Relation:
यदि a≥0, तो √(a2) = a।
उदाहरण:
√144 = 12
क्योंकि 122=144।
उदाहरण:
√625 = 25।
महत्वपूर्ण Observation:
General real number a के लिए √(a2)=|a|। उदाहरण: √((−5)2)=5।

10. Perfect-Square Factors द्वारा Square Root Simplify करना

मुख्य नियम:
√(a2b)=a√b, जहाँ a≥0 एवं b≥0।
उदाहरण:
√72
=√(36×2)
=6√2।
उदाहरण:
√200
=√(100×2)
=10√2।
Shortcut:
Root के अंदर सबसे बड़ा perfect-square factor खोजें। इससे answer एक ही step में simplest form में आ जाता है।
Scope Note:
Surds और rationalisation का detailed study Chapter 7 में किया जाएगा।

11. Like Roots का Addition एवं Subtraction

उदाहरण:
√50 + √8

=5√2+2√2
=7√2।
उदाहरण:
√75 − √27

=5√3−3√3
=2√3।
Exam Trap:
Generally √a+√b ≠ √(a+b)।

उदाहरण:
√9+√16=3+4=7,
जबकि √25=5।

12. Cube Roots एवं Perfect Cubes

Basic Relation:
∛(a3) = a.
उदाहरण:
∛216=6
क्योंकि 63=216।
उदाहरण:
∛(−125)=−5।
Observation:
Real cube root negative numbers के लिए भी defined होता है।

13. Perfect-Cube Factors द्वारा Cube Root Simplify करना

मुख्य नियम:
∛(a3b)=a∛b।
उदाहरण:
∛54
=∛(27×2)
=3∛2।
उदाहरण:
∛250
=∛(125×2)
=5∛2।
Shortcut:
8, 27, 64, 125, 216, 343, 512 और1000 जैसे perfect cubes जल्दी पहचानें।

14. Products एवं Fractions के Roots

Square-Root Rules:
a,b≥0 के लिए:
√(ab)=√a×√b

a≥0, b>0 के लिए:
√(a/b)=√a/√b
उदाहरण:
√(144/9)
=12/3
=4।
उदाहरण:
∛(64×125)
=4×5
=20।
सामान्य गलती:
Product rule का अर्थ यह नहीं है कि root को addition या subtraction पर distribute किया जा सकता है।

15. Decimal Perfect Squares के Square Roots

उदाहरण:
√1.44
=√(144/100)
=12/10
=1.2।
उदाहरण:
√0.0081
=√(81/10000)
=9/100
=0.09।
Shortcut:
Perfect-square decimal में decimal places की संख्या even हो तो temporarily decimal हटाएँ, integer का square root निकालें और answer में आधे decimal places रखें।
Exam Tip:
यह shortcut तभी use करें जब underlying integer perfect square हो।

16. Powers और Roots के साथ BODMAS

उदाहरण:
23 + √81 − ∛27

23=8
√81=9
∛27=3

अतः:
8+9−3=14।
उदाहरण:
(52−32) ÷ √16

25−9=16
√16=4

16÷4=4।
Exam Method:
Expression में सभी powers और roots पहले identify करें। Brackets के बाद इन्हें solve करके फिर ordinary operations पर जाएँ।

17. Fast Calculation के लिए Important Squares एवं Cubes

nn²nn³
1010011
1112128
12144327
13169464
141965125
152256216
162567343
172898512
183249729
19361101000
20400121728
25625153375
30900208000
Exam Shortcut:
Common squares और cubes याद होने पर root-finding method लगाने की आवश्यकता कई questions में नहीं पड़ती।

18. Common Exam Traps

Trap 1:
−a² और (−a)² को समान समझना।
Trap 2:
Power of a power में exponents को multiply करने के बजाय add करना।
Trap 3:
Negative exponent को reciprocal के बजाय negative number समझना।
Trap 4:
√(a+b)=√a+√b मान लेना।
Trap 5:
General real a के लिए √(a²)=|a| को भूल जाना।
Trap 6:
Long expression में powers एवं roots होने पर BODMAS ignore करना।
Trap 7:
Same-base exponent laws उपलब्ध होने के बावजूद large powers अलग-अलग calculate करना।

19. Quick Revision

एक नज़र में:
• Powers एवं roots BODMAS के Orders stage में solve होते हैं।
• am×an=am+n।
• am÷an=am−n, a≠0।
• (am)n=amn।
• a0=1, a≠0।
• a−n=1/an।
• √(a²)=|a|।
• Square root से perfect-square factors बाहर निकालें।
• Cube root से perfect-cube factors बाहर निकालें।
• केवल like radicals directly add/subtract होते हैं।
• √(a+b) generally √a+√b नहीं होता।
• Fast calculation के लिए common squares एवं cubes याद रखें।

20. Verified Previous-Year Questions

SSC MTS PYQ Squares & Square Root 3 मई 2023 · Shift II

प्रश्न 1. √[((132−82)×5)+4] का मान ज्ञात करें।

A. 23
B. 26
C. 24
D. 22
सही उत्तर: A. 23
132−82
=169−64
=105।

105×5+4
=529।

√529=23।
RRB Group D PYQ Square Root of Decimal 30 अगस्त 2022 · Shift III

प्रश्न 2. √1.0201 का मान ज्ञात करें।

A. 10.1
B. 101
C. 1.01
D. 1.02
सही उत्तर: C. 1.01
1.0201=10201/10000।

1012=10201 तथा 1002=10000।

इसलिए:
√1.0201
=101/100
=1.01।
RRB NTPC PYQ Cube Root 26 जुलाई 2021 · CBT-I · Shift I

प्रश्न 3. ∛150 किन दो consecutive integers के बीच स्थित है?

A. 6 और 7
B. 4 और 5
C. 7 और 8
D. 5 और 6
सही उत्तर: D. 5 और 6
53=125
63=216।

125<150<216।

इसलिए:
5<∛150<6।

अतः ∛150 5 और6 के बीच है।
RRB Group D PYQ Exponent Laws 5 फरवरी 2026 · Shift III

प्रश्न 4. यदि √[32 × 64−2 × (2/163) × 4m+3] = 2 है, तो m का मान क्या है?

A. 8
B. 7
C. 6
D. 4
सही उत्तर: B. 7
सभी terms को base 2 में लिखें:

32=25
64−2=2−12
2/163=2−11
4m+3=22m+6।

Root के अंदर:
25−12−11+2m+6
=22m−12।

Square root लेने पर:
2m−6=21।

m−6=1
m=7।

21. Practice MCQs

Practice MCQ

प्रश्न 1. सरल कीजिए: 23 + √81 − ∛27

A. 12
B. 14
C. 16
D. 18
सही उत्तर: B. 14
23=8, √81=9 तथा ∛27=3। इसलिए 8+9−3=14।
Practice MCQ

प्रश्न 2. सरल कीजिए: (52−32) ÷ √16

A. 2
B. 4
C. 6
D. 8
सही उत्तर: B. 4
25−9=16 तथा √16=4। इसलिए 16÷4=4।
Practice MCQ

प्रश्न 3. सरल कीजिए: √144 + ∛216

A. 16
B. 17
C. 18
D. 20
सही उत्तर: C. 18
√144=12 और ∛216=6। Sum=18।
Practice MCQ

प्रश्न 4. सरल कीजिए: 43 ÷ 24

A. 2
B. 4
C. 6
D. 8
सही उत्तर: B. 4
43=26। अतः 26÷24=22=4।
Practice MCQ

प्रश्न 5. सरल कीजिए: √(225/9)

A. 3
B. 5
C. 15
D. 25
सही उत्तर: B. 5
225/9=25 तथा √25=5।
Practice MCQ

प्रश्न 6. सरल कीजिए: (34 × 32) ÷ 35

A. 1
B. 3
C. 9
D. 27
सही उत्तर: B. 3
34+2−5=31=3।
Practice MCQ

प्रश्न 7. सरल कीजिए: √50 − √8

A. 2√2
B. 3√2
C. 5√2
D. 7√2
सही उत्तर: B. 3√2
√50=5√2 तथा √8=2√2। इसलिए 5√2−2√2=3√2।
Practice MCQ

प्रश्न 8. सरल कीजिए: ∛(64×125)

A. 10
B. 15
C. 20
D. 25
सही उत्तर: C. 20
∛64=4 तथा ∛125=5। इसलिए 4×5=20।
Practice MCQ

प्रश्न 9. सरल कीजिए: (√49)2 + ∛(−27)

A. 42
B. 44
C. 46
D. 52
सही उत्तर: C. 46
(√49)2=49 तथा ∛(−27)=−3। इसलिए 49−3=46।
Practice MCQ

प्रश्न 10. सरल कीजिए: 163/4

A. 4
B. 6
C. 8
D. 16
सही उत्तर: C. 8
163/4=(161/4)3=23=8।
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