CAT Exam > CAT Notes > Quantitative Aptitude (Quant) > Important Formulas: Surds and Indices
Important Formulas: Surds and Indices | Quantitative Aptitude (Quant) - CAT PDF Download
Surds and Indices form a crucial part of the Quantitative Aptitude section in competitive exams. These concepts help simplify complex expressions involving powers, roots, and irrational numbers. Mastering the underlying rules allows you to solve problems involving exponents, equations, and simplifications with speed and accuracy.
Laws of Indices
Indices (or exponents) represent repeated multiplication of the same number. Understanding their rules helps simplify expressions quickly.
When the bases are the same:
- aᵐ × aⁿ = aᵐ⁺ⁿ (Product Rule)
- aᵐ ÷ aⁿ = aᵐ⁻ⁿ (Quotient Rule)
- (aᵐ)ⁿ = aᵐⁿ (Power of a Power Rule)
- (a × b)ᵐ = aᵐ × bᵐ (Product of Bases Rule)
- (a ÷ b)ᵐ = aᵐ ÷ bᵐ (Quotient of Bases Rule)
- a⁰ = 1 (Zero Exponent Rule, valid when a ≠ 0)
- a⁻ⁿ = 1 ÷ aⁿ (Negative Exponent Rule)
Fractional and Rational Indices
A fractional index represents a root. Converting between roots and exponents makes it easier to handle radical expressions.
- a¹⁄ⁿ = nᵗʰ root of a = ⁿ√a
- aᵐ⁄ⁿ = (ⁿ√a)ᵐ = ⁿ√(aᵐ)
Important Relationships
These relationships help solve equations involving exponents quickly:
- If aˣ = aʸ ⇒ x = y (valid when a > 0 and a ≠ 1)
- If aˣ = bˣ ⇒ a = b (valid when x ≠ 0; and both a and b are positive and real numbers)
Surds: Key Concepts
A surd is an irrational root of a rational number that cannot be simplified into a rational form.
Examples: √2, ³√7, √5, etc.
Two surds √a and √b are like surds if a = b (e.g., 3√2 and 5√2 are like surds).
They are unlike surds if a ≠ b (e.g., √2 and √3).
Surds are used to express non-terminating, non-repeating decimal values in an exact form.
Basic Operations on Surds
The same arithmetic rules apply to surds, provided the radicands (numbers inside the roots) are positive.
√a × √b = √(a × b)
√a ÷ √b = √(a ÷ b)
(√a)² = a
ⁿ√a × ⁿ√b = ⁿ√(a × b)
Example: √3 × √12 = √36 = 6
Rationalisation of Denominators
Rationalisation is the process of removing a surd from the denominator of a fraction. Multiply the numerator and denominator by the conjugate of the denominator.
Examples:
1 ÷ (√a + √b) = (√a − √b) ÷ (a − b)
1 ÷ (√a − √b) = (√a + √b) ÷ (a − b)
This helps convert irrational denominators into rational ones for simplification.
Shortcut Techniques
(a) Convert all exponential terms to a common base to simplify comparisons or equations.
(b) Factorise the number under a root to extract perfect squares and simplify the surd.
Example:
√72 = √(36 × 2) = 6√2
√50 = √(25 × 2) = 5√2
Therefore, √72 + √50 = (6 + 5)√2 = 11√2
These techniques help in reducing lengthy calculations in CAT questions.
Common Simplifications
Certain forms appear frequently and can be memorised for faster computation:
√(a²) = |a| (the result is always positive)
√(a²b²) = ab
√(a/b) = √a ÷ √b
If √a + √b = 0 ⇒ a = b = 0
EduRev Tip: Surds and Indices questions often test your ability to simplify expressions and compare powers efficiently. Focus on base conversions, rationalisation, and identifying patterns in powers and roots. Practice regularly to build speed and accuracy.
The document Important Formulas: Surds and Indices | Quantitative Aptitude (Quant) - CAT is a part of the CAT Course Quantitative Aptitude (Quant).
All you need of CAT at this link: CAT
|
167 videos|229 docs|95 tests
|
FAQs on Important Formulas: Surds and Indices - Quantitative Aptitude (Quant) - CAT
| 1. What are the laws of indices and how do they apply in simplifying expressions? |  |
Ans. The laws of indices, also known as the laws of exponents, are rules that describe how to handle mathematical operations involving powers. Some key laws include:
1. a^m × a^n = a^(m+n): When multiplying two powers with the same base, you add the exponents.
2. a^m ÷ a^n = a^(m-n): When dividing two powers with the same base, you subtract the exponents.
3. (a^m)^n = a^(mn): When raising a power to another power, you multiply the exponents.
4. a^0 = 1: Any non-zero base raised to the power of zero equals one.
These laws are crucial for simplifying complex expressions in algebra and are frequently utilized in competitive exams.
| 2. How do fractional and rational indices work in mathematical expressions? | |
Ans. Fractional and rational indices represent roots in addition to powers. For instance, a^(1/n) is equivalent to the n-th root of a. Similarly, a^(m/n) can be interpreted as (n-th root of a) raised to the m-th power. This means that:
- a^(1/2) = √a (square root of a)
- a^(1/3) = ∛a (cube root of a)
- a^(m/n) = (√[n]{a})^m
Understanding these concepts allows for the simplification of expressions involving roots and powers, making it easier to solve problems in algebra.
| 3. What are surds and why are they important in mathematics? | |
Ans. Surds are irrational numbers that cannot be expressed as a simple fraction, typically represented as square roots, cube roots, etc., of non-perfect squares or cubes. Examples include √2, √3, and ∛5. Surds are significant in mathematics as they allow for precise representation of quantities that cannot be simplified into rational numbers. They are commonly encountered in geometry, algebra, and calculus, and understanding surds is essential for solving various mathematical problems.
| 4. How can I rationalize the denominator of a fraction involving surds? | |
Ans. Rationalizing the denominator involves eliminating surds from the denominator of a fraction. This can be done by multiplying the numerator and the denominator by a suitable form of the surd. For example, to rationalize 1/√a, multiply by √a/√a, resulting in (√a)/(a). In cases where the denominator involves a binomial with a surd, such as (√a + b), multiply by (√a - b)/(√a - b) to achieve a rational denominator. This technique is crucial for simplifying expressions and is often required in exams.
| 5. What are some shortcut techniques for simplifying expressions involving surds and indices? | |
Ans. There are several shortcut techniques that can be employed to simplify expressions involving surds and indices:
1. Use the laws of indices to combine powers and roots effectively.
2. Factor out common terms in surds to simplify them before performing operations.
3. For expressions like √a + √b, consider squaring the entire expression to eliminate the surds, if applicable.
4. Utilize conjugates to rationalize denominators quickly, as mentioned earlier.
5. Familiarize yourself with common surd values and their simplifications to speed up calculations. Practicing these techniques can significantly enhance efficiency in solving problems.
About this Document
Oct 19, 2025
Last updated
Document Description: Important Formulas: Surds and Indices for CAT 2025 is part of Quantitative Aptitude (Quant) preparation.
The notes and questions for Important Formulas: Surds and Indices have been prepared according to the CAT exam syllabus. Information about Important Formulas: Surds and Indices covers topics
like Laws of Indices, Fractional and Rational Indices, Important Relationships, Surds: Key Concepts, Basic Operations on Surds, Rationalisation of Denominators, Shortcut Techniques, Common Simplifications and Important Formulas: Surds and Indices Example, for CAT 2025 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Important Formulas: Surds and Indices.
Introduction of Important Formulas: Surds and Indices in English is available as part of our Quantitative Aptitude (Quant)
for CAT & Important Formulas: Surds and Indices in Hindi for Quantitative Aptitude (Quant) course.
Download more important topics related with notes, lectures and mock test series for CAT
Exam by signing up for free. CAT: Important Formulas: Surds and Indices | Quantitative Aptitude (Quant) - CAT
Description
Full syllabus notes, lecture & questions for Important Formulas: Surds and Indices | Quantitative Aptitude (Quant) - CAT - CAT | Plus excerises question with solution to help you revise complete syllabus for Quantitative Aptitude (Quant) | Best notes, free PDF download
Information about Important Formulas: Surds and Indices
In this doc you can find the meaning of Important Formulas: Surds and Indices defined & explained in the simplest way possible. Besides explaining types of
Important Formulas: Surds and Indices theory, EduRev gives you an ample number of questions to practice Important Formulas: Surds and Indices tests, examples and also practice CAT
tests
Related Searches
Important Formulas: Surds and Indices | Quantitative Aptitude (Quant) - CAT
,Free
,practice quizzes
,Sample Paper
,past year papers
,Important Formulas: Surds and Indices | Quantitative Aptitude (Quant) - CAT
,Semester Notes
,MCQs
,Extra Questions
,study material
,Previous Year Questions with Solutions
,Objective type Questions
,shortcuts and tricks
,Viva Questions
,Important Formulas: Surds and Indices | Quantitative Aptitude (Quant) - CAT
,mock tests for examination
,video lectures
,Important questions
,ppt
,Summary
,Exam
;
Additional Information about Important Formulas: Surds and Indices for CAT Preparation
Important Formulas: Surds and Indices Free PDF Download
The Important Formulas: Surds and Indices is an invaluable resource that delves deep into the core of the CAT exam.
These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective.
With the help of these notes, you can grasp complex subjects quickly, revise important points easily,
and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner,
allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material,
or toppers' notes, this PDF has got you covered. Download the Important Formulas: Surds and Indices now and kickstart your journey towards success in the CAT exam.
Importance of Important Formulas: Surds and Indices
The importance of Important Formulas: Surds and Indices cannot be overstated, especially for CAT aspirants.
This document holds the key to success in the CAT exam.
It offers a detailed understanding of the concept, providing invaluable insights into the topic.
By knowing the concepts well in advance, students can plan their preparation effectively.
Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.
Important Formulas: Surds and Indices Notes
Important Formulas: Surds and Indices Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to Important Formulas: Surds and Indices.
It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation.
Practice papers and question papers enable you to assess your progress effectively.
Additionally, the paper analysis provides valuable tips for tackling the exam strategically.
Access to Toppers' notes gives you an edge in understanding complex concepts.
Whether you're a beginner or aiming for advanced proficiency, Important Formulas: Surds and Indices Notes on EduRev are your ultimate resource for success.
Important Formulas: Surds and Indices CAT Questions
The "Important Formulas: Surds and Indices CAT Questions" guide is a valuable resource for all aspiring students preparing for the
CAT exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study Important Formulas: Surds and Indices on the App
Students of CAT can study Important Formulas: Surds and Indices alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Important Formulas: Surds and Indices,
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of Important Formulas: Surds and Indices is prepared as per the latest CAT syllabus.
|
© EduRev
|
Education Revolution
|
|
Signup on EduRev and stay on top of your study goals
10M+ students crushing their study goals daily