# 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

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.

#### Laws of Exponents

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.

#### Applications of Exponents and Powers

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.

#### Conclusion

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.

#### FAQs

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
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• Current Electricity
• Dual nature of Radiation and Matter
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• Gravitation
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• Wave Optics
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• Work, Power and Energy

### High School Chemistry

• Acids, Bases and Salts
• Alcohols, Phenols and Ethers
• Aldehydes, Ketones and Carboxylic Acids
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### High School Biology

• Absorption and Movement of Water in Plants
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### High School Math

• Algebra – Arithmatic Progressions
• Algebra – Complex Numbers and Quadratic Equations
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• Algebra – Pair of Linear Equations in Two Variables
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• Algebra – Principle of Mathematical Induction
• Binomial Theorem
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• Permutations and Combinations
• Probability
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• Trignometry – Height and Distance
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### 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
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• Coal And Petroleum
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• Conservation Of Plants And Animals
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• Elements and Compounds
• Fibre To Fabric
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• Force And Pressure
• Forests: Our Lifeline
• Friction
• Fun With Magnets
• Garbage In, Garbage Out
• Getting To Know Plants
• Health and Hygiene
• Heat
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• Life Processes: Nutrition in Animals and Plants
• Materials: Metals And Non-Metals
• Matter and Its States
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• Our Environment
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• Reaching The Age Of Adolescence
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• The Living Organisms And Their Surroundings
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• Universe
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• Water: A Precious Resource
• Weather, Climate And Adaptations Of Animals To Climate
• Winds, Storms And Cyclones

### Middle School Math

• 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
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• 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