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Chapter 13 – Exponents and Powers

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

Introduction

Exponents and Powers is an important NCERT maths chapter that explains the basics of exponential functions and their importance. You must have learned the concept of exponents and power in your basic algebra class, where you use powers over variables. For example, to simplify (a + b)2 = a2 + 2ab + b2.

We have various laws on exponents and powers that facilitate the simplification of these numbers. Let’s check the general exponential form at present.

The general form of exponents is written as (a)n, where a and n are any unique natural numbers.

The expanding of (a)n = a x a x a x a x a x a x……..n times.
For example, (2)4 = 2 x 2 x 2 x 2 = 16
The (a)n is the shorthand notation of a very large number that has been written in a simple form.

Let’s check some basic laws of exponents that apply to the exponent and power numbers.

  1. Multiplication Law: When we multiply powers with the same base, we need to add the different powers of the base. This can be depicted mathematically as:
    (a)m x (a)n = (a)(m+n)
    Where a, m, and n are natural numbers.
  2. Division Law: When we divide powers with the same base, we need to subtract the denominator’s base power from the numerator’s base. This can be depicted mathematically as:
    am ÷ an = am / an = a(m-n)
    Where a, m, and n are any natural numbers.
  3. Negative Exponent Law: It is represented as:
    a(-m)= 1/(a)m
    Where a and m are any integer.

Apart from these three basic laws, we have some exponent and power rules that come into play. We will check them below.

  • Rule 1: a0 = 1. This rule is a basic hypothesis that can be proved by using the division law discussed above.
  • Rule 2: ((a)m)n = (a)(m x n). Multiplying powers with the same exponents, we get an equivalent exponent to multiplicate individual powers. We can also call it taking the power of a power.
  • Rule 3: am/bm = (a/b)m. Dividing powers with the same exponents. When we have a division with the same exponents and same base, we can simplify it, as shown in the rule above.
  • Rule 4: am × bm = (ab)m. When we multiplicate with the same exponents and the same base, we can simplify it, as shown in the rule above.

Numbers and exponents are the basics of mathematics and are used to depict nature’s physical quantities. There are many usages of exponents and powers in different fields and industries. Some of them are given below.

    • They are used extensively in the decimal number system to depict a scientific notation of large numbers.
    • They are also used in Physics to solve formulas and derive their proofs.
    • They have importance in Chemistry to note the reaction time of any chemical reaction.
    • Moreover, they find application in Geometrical Algebra and Coordinate System.

Exponents and powers are an important mathematical topic used extensively in both maths and science. You must have started learning them in the algebra section and while solving proofs in the physics section. Apart from this, they are also used in chemistry and computer science. There are some simple rules and laws associated with exponents and powers that simplify their use. The rules and laws are discussed above in detail.

  1. What are exponents and powers?
    Exponents and powers are simplified versions of huge numbers in a fixed format. We use exponents to write very large numbers in a simplified way.
  2. What is the power of a power rule for exponents?
    The power of a power rule states that when a certain variable or constant is double powered, both the power superscripts multiply to give a more simplified version. For example, ((2)2)3 = (2)(2*3) = (2)6
  3. What are the five rules of exponents?
    The five rules of exponents tell us about the different rules associated with the calculation of the exponents. They have been explained above.
  4. How do you teach exponents and powers?
    Teaching exponents and powers are simple as they don’t require the knowledge of any complicated concepts. Five simple exponent rules need to be understood before beginning to learn the concept of exponents and powers.
  5. How do you calculate exponential powers?
    We must multiply the base number with the exponent number to get the solution. We can also use the logarithmic function to deal with complex exponent functions.
  6. What are the rules of exponents?
    The rules of exponents tell us about the different rules associated with the calculation of the exponents. They are explained below.
    • Multiplication Law: When we multiply powers with the same base, we need to add the different powers of the base numbers. This can be depicted mathematically as:(a)m x (a)n = (a)(m+n)
      Where a, m, and n are natural numbers.
    • Division Law: When we divide powers with the same base, we need to subtract the denominator’s base power from the numerator’s base. This can be depicted mathematically as:
      am ÷ an = am / an = a(m-n)
      Where a, m, and n are any natural numbers.
    • Negative Exponent Law: It is represented as:
      a(-m) = 1/(a)m
      Where a and m are any integer.

To get a thorough understanding of Exponents and Powers, download MSVgo. It is an app that contains video tutorials to help you understand even the toughest of the topics.

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
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  • Metals And Non-Metals
  • Mole Concept and Stoichiometry
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  • 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
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  • 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
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