Syllabus covered in the MSVgo app

Download MSVgo app now!

Chapter 6 – Squares and Square Roots

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

Introduction

You must have come across real-life scenarios when squares and square roots are used in the form of the Pythagorean theorem by architects, engineers and carpenters. This forms an important chapter for students from an exam point of view as it is used in different chapters in higher education. Let us see more details about squares and square roots, their properties, and how to find these in case of a whole number and decimal in this article.

Introduction to Squares and Square Roots

Square

Square of any number is the number multiplied by itself. For example, for finding the square of a number ‘z’, we will multiply z × z. It is also represented as z2.

Example:

Square of 40, i.e., 402 = 1600

Similarly, let us see square roots of decimals:

2.1, i.e.,  2.212 = 4.41

Now, square a negative number:

(-4)2 = 16

A negative number squared always results in a positive number. Finding the square of a number in case of double digit or higher numbers may be cumbersome; you can use the following method:

272 = (20 + 7)2 = 202 + 72 + 2(20 × 7) = 729

Square Roots

The square root of any number is the value which when multiplied with itself results in the original number.

In other words, if the square root of z is a, then a2 = z.

We write the square root symbol as √, so the above can be written as √z = a.

To be specific, square roots of a number always result in 2 values, one positive and another negative, hence:

√z = +a and -a.

Let us see a numerical example:

√225 = 25 (thus, 252 = 225)

Similarly, in case of a decimal number:

√0.16 = 0.4

i. Perfect squares: When whole numbers are squared, it results in perfect squares.

Example: 82 = 16.

ii. The squares of numbers always end with 0, 1, 4, 5, 6, or 9. You will never find the square ending with 2, 3, 7, or 8 as in the following cases:

NumberSquare
11
24
39
416
525
636
749
864
981
10100
11121
12144
13169
14196
15225
16256
17289
18324
19361
20400

iii. If numbers are ending with 1 or 9, i.e., if these digits are in the unit’s place, then their square will end with 1.

NumberSquare
11
981
11121
19361
21441

iv. The square of numbers with 4 or 6 in the unit’s place always ends with 6.

NumberSquare
416
636
14196
16256

v. If ‘n’ is a natural number and ‘n + 1’ is the next natural number then:
(n + 1)2 – n2 = (n2 + 2n + 1) – n2 = 2n + 1.
So, we can tell that between the squares of 2 consecutive numbers, there are 2n non-perfect square numbers.
Example: How many numbers are there between 42 and 52?
Here, n = 4 and n + 1 = 5
So there are (2 × 4) = 8 non square numbers between 42 and 52, i.e., 17, 18, 19, 20, 21, 22, 23, 24.

vi. A natural number is not a perfect square if you can’t express it as a sum of successive numbers beginning from 1. This can be used for finding if a number is a perfect square. In other words, for knowing if you need to check if n2 is a perfect square, add n numbers from 1 and see if the result is n2.
Example: 62 = 36
So let us see the sum of 1st 6 odd numbers, which is = 1 + 3 + 5 + 7 + 9 + 11 = 36.

vii. Square of an odd number is the sum of two consecutive positive integers.
Example: 92 = 81, which can be written as 40 + 41.

viii. The product of two consecutive odd or even numbers can be written as n2 – 1, where ‘n’ is a natural number.
We can also write (n + 1)(n – 1) = n2 – 1.
Example: Let n = 5, so 6 × 4 = 24 = 25 – 1.

ix. Pythagorean triplets:
If we take any natural number ‘n’ greater than 1, then:
(2n)2 + (n2 – 1)2 = (n2 + 1)2
Here, 2n, n2 – 1 and n2 + 1 are termed as pythagorean triplets.
Example:
Let n = 3
So, (2 * 3)2 + (32 – 1)2 = (32 + 1)2
= 36 + 64 = 100 = 102
i.e., 62 + 82 = 102
Hence, 6, 8, and 10 are Pythagorean triplets.

i. Repeated subtraction method

We have already seen that every perfect square can be written as the sum of successive odd natural numbers beginning from one.

Let us consider an example 64 and go in the reverse order, i.e., subtract the successive odd numbers from the given number, starting from 1 till you get a zero. If you get zero at the nth step, then the square root of the given number is ‘n’.

So, 64 – 1 = 63

63 – 3 = 60

60 – 5 = 55

55 – 7 = 48

48 – 9 = 39

39 – 11 = 28

28 – 13 = 15

15 – 15 = 0

It took us 8 steps to reach zero, so √64 = 8.

ii. Prime factorisation

We will understand this by an example:

Let us take a number 324. Its prime factorisation comes to 2 × 2 × 3 × 3 × 3 × 3. We can pair the prime factors, which gives 22 x 32 x  32.

Hence, we can tell that square root of 324 is:
324 = 2 × 3 × 3 = 18

What if the prime factors are not in pairs?
Example: 48 = 2 × 2 × 2 × 2 × 3.
Here, since we are unable to form pairs, this is not a perfect square.

iii. Division method

The prime factorisation method can be time taking in case of larger numbers. Hence the division method can be used in such cases.

We will understand this by an example: 529.

Put a bar on the pair of digits from the extreme right end. In case there are an odd number of digits, a single digit will be left in the left end without a bar. 3 will be without a bar in our case.

Take the largest number whose square is less than that of the number under the extreme left bar. So it will be 22 < 5 < 32.

And we will take 2 as the divisor, and the number under the extreme left bar will be the quotient. You will get reminder 1 after dividing.

Then the number under the next bar, i.e., 29, has to be brought down right to the existing reminder, which forms 129. This has to be divided by 4 (doubling the divisor).

Now the new digit in the quotient will be the largest possible number to fill the blank and it will adhere to the formula of new divisor multiplied to new quotient, which results in a number less than or equal to the dividend.

Here, it will be 42 x 4, i.e., 84. Since 43 x 3 = 129, the new digit chosen is 3 and the remainder is obtained which is 0. So, the process ends, and 529 = 23.

What is the relationship between square roots and squares?

The square root of a number is the number which when multiplied with itself results in the original number.

So, if z2 = a,

Then, √a = z.

Does a square root cancel out a square?

Yes, when the number is a perfect square, a square root will cancel out the square.

How do you simplify square roots with square roots?

Consider an example to understand this: √64 = √82.

Here, square root will cancel out the square, so the answer will be 8.

Does a negative have a square root?

No. You can find the square root of non-negative numbers only.

This was a brief explanation of various aspects of squares and square roots. You can refer to MSVgo app or website for learning the topic in a much better way. It has numerous examples along with visual illustrations to clarify concepts so that you don’t face issues while solving even the most complex problems. Once the concept is clear, practice the different types of questions well to get a good grasp on the topic.

High School Physics

  • Alternating Current
  • Atoms
  • Communication Systems
  • Current Electricity
  • Dual nature of Radiation and Matter
  • Electric Charges and Fields
  • Electricity
  • Electromagnetic Induction
  • Electromagnetic Waves
  • Electron Beams and Radioactivity
  • Electrons and Photons
  • Electrostatic Potential and Capacitance
  • Fluid Pressure
  • Force and Acceleration
  • Force And Laws Of Motion
  • Gravitation
  • Internal Energy
  • Kinetic Theory
  • Law of motion
  • Light – Reflection And Refraction
  • Magnetic Effects Of Electric Current
  • Magnetism and Matter
  • Management Of Natural Resources
  • Mechanical properties of Fluids
  • Mechanical properties of Solids
  • Motion
  • Motion in a plane
  • Motion in a straight line
  • Moving Charges and Magnetism
  • Nuclear Energy
  • Nuclei
  • Oscillations
  • Our Environment
  • Paths of Heat
  • Physical world
  • Ray optics and optical instruments
  • Semiconductor Devices
  • Semiconductor Electronics: Materials, Devices and Simple Circuits
  • Simple Machines
  • Sound
  • Sources Of Energy
  • Specific and Latent Heats
  • Spherical Mirrors
  • Static Electricity
  • Systems of Particles and Rotational motion
  • Thermal properties of matter
  • Thermodynamics
  • Units and Measurement
  • Vectors, Scalar Quantities and Elementary Calculus
  • Wave Optics
  • Waves
  • Work, Power and Energy

High School Chemistry

  • Acids, Bases and Salts
  • Alcohols, Phenols and Ethers
  • Aldehydes, Ketones and Carboxylic Acids
  • Aliphatic and Aromatic Hydrocarbons
  • Alkyl and Aryl Halides
  • Amines
  • Analytical Chemistry 
  • Atomic Structure
  • Atoms And Molecules
  • Basic concepts of Chemistry
  • Biomolecules
  • Carbon And Its Compounds
  • Carboxylic acids and Acid Derivatives
  • Chemical Bonding and Molecular Structures
  • Chemical Energetics
  • Chemical Equilibria
  • Chemical Kinetics
  • Chemical Reactions And Equations
  • Chemical Reactions and Their Mechanisms
  • Chemistry in Everyday Life
  • Chemistry of p-Block elements
  • Chemistry of Transition and Inner Transition
  • Classification of Elements
  • Coordination Compounds
  • Cyanide, Isocyanide, Nitro compounds and Amines
  • Electrochemistry
  • Electrolysis
  • Elements, Compounds and Mixtures
  • Environmental Chemistry
  • Equilibrium
  • Ethers and Carbonyl compounds
  • Haloalkanes and Haloarenes
  • Hydrocarbons
  • Hydrogen
  • Ideal solutions
  • Introduction to Organic Chemistry
  • Ionic equilibria
  • Matter
  • Matter Around Us
  • Matter In Our Surroundings
  • Metallurgy
  • Metals And Non-Metals
  • Mole Concept and Stoichiometry
  • Natural Resources
  • Organic Chemistry – Basic Principles
  • Periodic Classification of Elements
  • Physical and Chemical Changes
  • Physical and Chemical Properties of Water
  • Polymers
  • Preparation, Properties and Uses of Compounds
  • Principles and Processes of Isolation of Elements
  • Redox Reactions
  • Relative Molecular Mass and Mole
  • States of Matter
  • Structure Of The Atom
  • Study of Compounds
  • Study of Gas Laws
  • Study of Representative Elements
  • Surface Chemistry
  • The d-block and f-block elements
  • The Gaseous State
  • The p-Block Elements
  • The Periodic Table
  • The s-Block Elements
  • The Solid State
  • Thermodynamics

High School Biology

  • Absorption and Movement of Water in Plants
  • Adolescent Issues
  • Anatomy of Flowering Plants
  • Animal Kingdom
  • Bacteria and Fungi-Friends and Foe
  • Biodiversity and Conservation
  • Biofertilizers
  • Biological Classification
  • Biomedical Engineering
  • Biomolecules
  • Biotechnology and its Applications
  • Biotic Community
  • Body Fluids and Circulation
  • Breathing and Exchange of Gases
  • Cell – Unit of Life
  • Cell Cycle and Cell Division
  • Cell Division and Structure of Chromosomes
  • Cell Reproduction
  • Cellular Respiration
  • Chemical Coordination and Integration
  • Circulation
  • Control And Coordination
  • Crop Improvement
  • Digestion and Absorption
  • Diversity In Living Organisms
  • Ecosystem
  • Environmental Issues
  • Excretory Products and their Elimination
  • Flowering Plants
  • Genes and Chromosomes
  • Health and Diseases
  • Health and Its Significance
  • Heredity And Evolution
  • Heredity and Variation
  • How Do Organisms Reproduce?
  • Human Diseases
  • Human Eye And Colourful World
  • Human Health and Disease
  • Human Population
  • Human Reproduction
  • Hygiene
  • Improvement In Food Resources
  • Integumentary System- Skin
  • Kingdom Fungi
  • Kingdom Monera
  • Kingdom Protista
  • Life Processes
  • Locomotion and Movement
  • Microbes in Human Welfare
  • Mineral Nutrition
  • Molecular Basis of Inheritance
  • Morphology of Flowering Plants
  • Neural Control And Coordination
  • Nutrition in Human Beings
  • Organism and Population
  • Photosynthesis
  • Photosynthesis in Higher Plants
  • Plant Growth and Development
  • Plant Kingdom
  • Pollination and Fertilization
  • Pollution; Sources and its effects
  • Principles of Inheritance and Variation
  • Reproduction and Development in Angiosperms
  • Reproduction in Organisms
  • Reproductive Health
  • Respiration in Human Beings
  • Respiration in Plants
  • Respiratory System
  • Sexual Reproduction in Flowering Plants
  • Strategies for Enhancement in Food Production
  • Structural Organisation in Animals
  • Structural Organisation of the Cell
  • The Endocrine System
  • The Fundamental Unit Of Life
  • The Living World
  • The Nervous System and Sense Organs
  • Tissues
  • Transpiration
  • Transport in Plants

High School Math

  • Algebra – Arithmatic Progressions
  • Algebra – Complex Numbers and Quadratic Equations
  • Algebra – Linear Inequalities
  • Algebra – Pair of Linear Equations in Two Variables
  • Algebra – Polynomials
  • Algebra – Principle of Mathematical Induction
  • Algebra – Quadratic Equations
  • Binomial Theorem
  • Calculus – Applications of Derivatives
  • Calculus – Applications of the Integrals
  • Calculus – Continuity and Differentiability
  • Calculus – Differential Equations
  • Calculus – Integrals
  • Geometry – Area
  • Geometry – Circles
  • Geometry – Conic Sections
  • Geometry – Constructions
  • Geometry – Introduction to Euclid’s Geometry
  • Geometry – Three-dimensional Geometry
  • Geometry – Lines and Angles
  • Geometry – Quadrilaterals
  • Geometry – Straight Lines
  • Geometry – Triangles
  • Linear Programming
  • Matrices and Determinants
  • Mensuration – Areas
  • Mensuration – Surface Areas and Volumes
  • Number Systems
  • Number Systems – Real Numbers
  • Permutations and Combinations
  • Probability
  • Sequence and Series
  • Sets and Functions
  • Statistics 
  • Trignometry – Height and Distance
  • Trignometry – Identities
  • Trignometry – Introduction

Middle School Science

  • Acids, Bases And Salts
  • Air and Its Constituents
  • Basic Biology
  • Body Movements
  • Carbon and Its Compounds
  • Cell – Structure And Functions
  • Changes Around Us
  • Chemical Effects Of Electric Current
  • Chemistry in Your Life
  • Coal And Petroleum
  • Combustion And Flame
  • Components Of Food
  • Conservation Of Plants And Animals
  • Crop Production And Management
  • Electric Current And Its Effects
  • Electricity And Circuits
  • Elements and Compounds
  • Fibre To Fabric
  • Food production and management
  • Force And Pressure
  • Forests: Our Lifeline
  • Friction
  • Fun With Magnets
  • Garbage In, Garbage Out
  • Getting To Know Plants
  • Health and Hygiene
  • Heat
  • Hydrogen
  • Life Processes: Nutrition in Animals and Plants
  • Light, Shadows And Reflections
  • Materials: Metals And Non-Metals
  • Matter and Its States
  • Metals and Non-metals
  • Micro Organisms: Friend And Foe
  • Motion And Measurement Of Distances
  • Motion And Time
  • Nutrition In Animals
  • Nutrition In Plants
  • Organization in Living Things
  • Our Environment
  • Physical And Chemical Changes
  • Pollution and conservation
  • Pollution Of Air And Water
  • Reaching The Age Of Adolescence
  • Reproduction In Animals
  • Reproduction In Plants
  • Respiration In Organisms
  • Rocks and Minerals
  • Separation Of Substances
  • Simple Machines
  • Soil
  • Some Natural Phenomena
  • Sorting Materials Into Groups
  • Sound
  • Stars And The Solar System
  • Structure of Atom
  • Synthetic Fibers And Plastics
  • The Living Organisms And Their Surroundings
  • Transfer of Heat
  • Transformation of Substances
  • Transportation In Animals And Plants
  • Universe
  • Waste-water Story
  • Water: A Precious Resource
  • Weather, Climate And Adaptations Of Animals To Climate
  • Winds, Storms And Cyclones

Middle School Math

  • Addition
  • Area and Its Boundary
  • Boxes and Sketches
  • Data Handling
  • Fun With Numbers
  • Heavy and Light
  • How Many
  • Long And Short
  • Mapping
  • Measurement
  • Money
  • Multiplication and Factors
  • Multiply and Divide
  • Numbers
  • Parts and Wholes
  • Pattern Recognition
  • Patterns
  • Play With Patterns
  • Rupees And Paise
  • Shapes And Angles
  • Shapes And Designs
  • Shapes and Space
  • Similarity
  • Smart Charts
  • Squares
  • Subtraction
  • Tables And Shares
  • Tenths and Hundredths
  • Time
Please switch to portrait mode
for the best experience.
Click to open Popup