CAT 2025 Lesson : Surds & Indices - Concepts & Cheatsheet

12. Cheatsheet

Note: The video for this module contains a summary of all the concepts covered in this lesson. The video would serve as a good revision. Please watch this video in intervals of a few weeks so that you do not forget the concepts. Below is a cheatsheet that includes all the formulae but not necessarily the concepts covered in the video.

1) As

(a + b)(a - b) = a^2 - b^2

, for the term

a + b

, the conjugate is

a - b

and vice versa.

2) To compare two surds, you can

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(1) Subtract/add the same integer to both surds to make them comparable

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(2) Square the surds if the numbers are close after step 1.

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(3) To compare square roots, find their integral ranges.

3) To compare square roots, we can find the range of the square root and conclude if a particular range is completely outside of another.

4) To find the square root of a quadratic surd, we express the surd in the form of

a^2 + b^2 + 2ab

\space a^2 + b^2

forms the rational part and

2ab

forms the irrational part.

5) Indices Rules

S.No.	Rule
1	$a^0 = 1$
2	$a^1 = a$
3	$1^n = 1$
4	$a^m \times a^n = a^{m + n}$
5	$\dfrac{a^m}{a^n} = a^{m - n}$
6	$a^{-n} = \dfrac{1}{a^n}$
7	$({a^m})^n = a^{m \times n}$
8	${{a^m}^{n}} = \large{a}^ {({m}^{n})}$
9	$a^n \times b^n = (ab)^n$
10	$\dfrac{{a}^n}{{b}^n} = \left( \dfrac {a}{b} \right)^n$
11	${a}^\frac{m}{n} = \sqrt[n]{a^m}$

6) Comparing Indices

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When the powers are integers, divide each of the powers by their HCF and compare.

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When the powers are fractions, multiply each of the powers by the LCM of denominators and compare.

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