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Surds & Indices

Surds And Indices

MODULES

Basics of Surds
Comparison of Surds
Root of Surds
Indices Rules
Comparing Indices
Past Questions

CONCEPTS & CHEATSHEET

Concept Revision Video

SPEED CONCEPTS

Surds & Indices 1
-/10

PRACTICE

Surds & Indices : Level 1
Surds & Indices : Level 2
Surds & Indices : Level 3
ALL MODULES

CAT 2025 Lesson : Surds & Indices - Indices Rules

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2. Indices

Numbers or variables written in the form
aba^bab means a needs to be multiplied b times.
an=a×a×a×...n timesa^n = a \times a \times a \times ..._{n\ times}an=a×a×a×...n times​
∴34=3×3×3×3=81\therefore 3^4 = 3 \times 3 \times 3 \times 3 = 81∴34=3×3×3×3=81

In
aba^bab, aaa is called the base and bbb is called the power, exponent or index.
(Indices is the plural for index)

The following are the rules for indices

S.No. Rule Example
1 a0=1a^0 = 1a0=1 30=240=(−137)0=13^0 = 24^0 = (-137)^{0} = 130=240=(−137)0=1
2 a1=aa^1 = aa1=a 41=4,751=754^1 = 4, 75^1 = 7541=4,751=75
3 1n=11^n = 11n=1 15=1795=1−15=11^5 = 1^{795} = 1^{-15} = 115=1795=1−15=1
4 am×an=am+na^m \times a^n = a^{m + n}am×an=am+n 35×37=35+7=3123^5 \times 3^7 = 3^{5 + 7} = 3^{12}35×37=35+7=312
5 aman=am−n\dfrac{a^m}{a^n} = a^{m-n}anam​=am−n 3537=35−7=3−2\dfrac{3^5}{3^7} = 3^{5 - 7} = 3^{-2}3735​=35−7=3−2
6 a−n=1ana^{-n} = \dfrac{1}{a^n}a−n=an1​ 3−2=132=193^{-2} = \dfrac{1}{3^2} = \dfrac{1}{9}3−2=321​=91​
7 (am)n=am×n({a^m})^n = a^{m \times n}(am)n=am×n (53)2=53×2=56({5^3})^2 = 5^{3 \times 2} = 5^6(53)2=53×2=56
8 amn=a(mn)\large{{a^m}^{n}} = \large{a}^ {({m}^{n})}amn=a(mn) 532=59\large{{5^3}^2} = 5^9532=59
9 an×bn=(ab)na^n \times b^n = ({ab})^nan×bn=(ab)n 138×28=26813^8 \times 2^8 = 26^8138×28=268
10 anbn=(ab)n\dfrac{{a}^n}{{b}^n} = \left( \dfrac {a}{b} \right)^nbnan​=(ba​)n 13828=(132)8\dfrac{{13}^8}{{2}^8} = \left( \dfrac {13}{2} \right)^828138​=(213​)8
11 amn=amn{a}^\frac{m}{n} = \sqrt[n]{a^m}anm​=nam​ 543=543{5}^\frac{4}{3} = \sqrt[3]{5^4}534​=354​

To extend rule
8\bm{8}8, if the powers are raised one over the other then, we start with applying the top most power and then move downwards.

3232=329=3512{{3^2}^3}^2 = {3^2}^9 = 3^{512}3232=329=3512

If brackets/parentheses would have been used, then the answer would have been different. We would apply rule
7\bm{7}7.

((32)3)2=32×3×2=312((3^2)^3)^2 = 3^{2 \times 3 \times 2} = 3^{12}((32)3)2=32×3×2=312

Example 6

Simplify 823×53−(23)2÷38^\frac{2}{3} \times 5^3 - ( 2^3 )^2 \div 3832​×53−(23)2÷3

Solution

=(82)13×125−263= ( 8^2 )^\frac{1}{3} \times 125 - \dfrac{2^6}{3}=(82)31​×125−326​

=(64)13×125−643=(43)13×125−643= ( 64 )^\frac{1}{3} \times 125 - \dfrac{64}{3} = ( 4^3 )^{\frac{1}{3}} \times 125 - \dfrac{64}{3}=(64)31​×125−364​=(43)31​×125−364​

=4×125−643= 4 \times 125 - \dfrac{64}{3}=4×125−364​ =500−643= 500 - \dfrac{64}{3}=500−364​ =14363= \dfrac{1436}{3}=31436​

Answer:
14363\dfrac{1436}{3}31436​

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