Exponents and Powers: Class 7 Maths NCERT Guide

Have you ever tried to write down a very, very large number, like the mass of the Earth? It has so many zeros! It's about 5,972,000,000,000,000,000,000,000 kg. Writing and reading such numbers is difficult and confusing. This is where exponents and powers come to the rescue! They are a super-convenient mathematical shorthand for writing repeated multiplication. Instead of writing 10 × 10 × 10, we can simply write 10³. In this chapter, you will learn what exponents and powers are, understand the fundamental 'laws of exponents' that make calculations much faster, and practice simplifying complex expressions. By the end of this guide, you will be able to handle very large and very small numbers with confidence, a skill that is essential in science and higher mathematics.

Understanding Exponents and Powers

Exponents are a way of expressing that a number is being multiplied by itself a certain number of times. Let's take the number 10,000. We can write it as 10 × 10 × 10 × 10. This is a bit long. Using exponents, we can write this as 10⁴. Here, the number 10 is called the base, and the small number 4 written on the top right is called the exponent (or power or index). The entire expression 10⁴ is read as '10 raised to the power of 4' or 'the 4th power of 10'. The exponent simply tells you how many times to write down the base and multiply it.

Let's look at another example: 2⁵.
Here, the base is 2 and the exponent is 5.
This means we need to multiply 2 by itself 5 times:
2⁵ = 2 × 2 × 2 × 2 × 2 = 32.
So, 2⁵ is the 'exponential form' and 32 is its value. This simple idea is the foundation for working with very large numbers easily.

Key Terminology in Exponents and Powers

Base
The number that gets multiplied by itself. In 8³, the base is 8.
Exponent
The small number written in the upper-right corner of the base. It indicates how many times the base is multiplied by itself. In 8³, the exponent is 3.
Power
The entire expression, including the base and exponent (e.g., 8³). It can also refer to the result of the calculation. 'The third power of 8' is 512.
Exponential Form
A compact way of writing repeated multiplication using a base and an exponent, like 5⁴ instead of 5 × 5 × 5 × 5.

The Fundamental Laws of Exponents

  1. Law 1: Multiplying Powers with the Same Base — Rule: aᵐ × aⁿ = aᵐ⁺ⁿ When you multiply two powers that have the same base, you can just add their exponents. For example, 5² × 5³ = 5²⁺³ = 5⁵.
  2. Law 2: Dividing Powers with the Same Base — Rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ When you divide two powers with the same base, you subtract the exponents. For example, 7⁶ ÷ 7² = 7⁶⁻² = 7⁴.
  3. Law 3: Taking a Power of a Power — Rule: (aᵐ)ⁿ = aᵐˣⁿ When an exponential expression is raised to another power, you multiply the exponents. For example, (3⁴)² = 3⁴ˣ² = 3⁸.
  4. Law 4: Multiplying Powers with the Same Exponent — Rule: aᵐ × bᵐ = (a × b)ᵐ When you multiply powers with different bases but the same exponent, you multiply the bases and keep the exponent. For example, 2⁴ × 3⁴ = (2 × 3)⁴ = 6⁴.
  5. Law 5: Dividing Powers with the Same Exponent — Rule: aᵐ ÷ bᵐ = (a / b)ᵐ When you divide powers with different bases but the same exponent, you can divide the bases and keep the exponent. For example, 10³ ÷ 5³ = (10/5)³ = 2³.
  6. Law 6: The Zero Exponent — Rule: a⁰ = 1 (for any non-zero number 'a') Any number (except zero) raised to the power of zero is always equal to 1. For example, 29⁰ = 1, and (-15)⁰ = 1.

Applying the Laws - Worked Examples

  • Example 1: Simplify 4³ × 4⁵ Solution: We are multiplying powers with the same base (4). Step 1: Identify the law to use: aᵐ × aⁿ = aᵐ⁺ⁿ. Step 2: Add the exponents: 3 + 5 = 8. Step 3: Keep the base and use the new exponent. Final Answer: 4⁸
  • Example 2: Simplify (6⁵)² Solution: This is a case of a power raised to another power. Step 1: Identify the law to use: (aᵐ)ⁿ = aᵐˣⁿ. Step 2: Multiply the exponents: 5 × 2 = 10. Step 3: Keep the base and use the new exponent. Final Answer: 6¹⁰
  • Example 3: Simplify and write in exponential form: (5¹⁰ ÷ 5⁶) × 5³ Solution: First, we simplify the expression inside the brackets. Step 1 (Brackets): Use the division law aᵐ ÷ aⁿ = aᵐ⁻ⁿ. So, 5¹⁰ ÷ 5⁶ = 5¹⁰⁻⁶ = 5⁴. Step 2 (Multiplication): The expression becomes 5⁴ × 5³. Now use the multiplication law aᵐ × aⁿ = aᵐ⁺ⁿ. So, 5⁴ × 5³ = 5⁴⁺³ = 5⁷. Final Answer: 5⁷

Important Tips for Exams

Here are a few common points where students make mistakes:

  • Mixing up Bases: Do not add or multiply the bases when using the first two laws. For example, 3² × 3⁴ is 3⁶, NOT 9⁶.
  • Confusing Exponent Addition and Multiplication: Remember, aᵐ × aⁿ = aᵐ⁺ⁿ (add exponents), but (aᵐ)ⁿ = aᵐˣⁿ (multiply exponents). Keep these two rules separate in your mind.
  • The Zero Power: Don't forget that any non-zero number raised to the power of 0 is 1. For example, (5 × 12)⁰ = 1. It is not 0!
  • Negative Bases: Pay attention to brackets with negative bases. (-3)² = (-3) × (-3) = 9 (positive). But (-3)³ = (-3) × (-3) × (-3) = -27 (negative). An even exponent on a negative base gives a positive result, while an odd exponent gives a negative result.

Practice Questions with Solutions

  • Q: Express 512 as a power of 2. A: Step 1: To express 512 as a power of 2, we use prime factorization and repeatedly divide by 2. 512 ÷ 2 = 256 256 ÷ 2 = 128 128 ÷ 2 = 64 64 ÷ 2 = 32 32 ÷ 2 = 16 16 ÷ 2 = 8 8 ÷ 2 = 4 4 ÷ 2 = 2 2 ÷ 2 = 1 Step 2: Count how many times we divided by 2. We can see that 2 is multiplied by itself 9 times. Step 3: Write this in exponential form. Final answer: 2⁹
  • Q: Simplify and write in exponential form: [ (3⁷)² × 5³ ] / [ 9² × 5² ] A: Step 1: Simplify each part of the expression. First, express 9 as a power of 3. 9 = 3². So, 9² = (3²)². Step 2: Apply the power of a power rule (aᵐ)ⁿ = aᵐˣⁿ to both (3⁷)² and (3²)². (3⁷)² = 3⁷ˣ² = 3¹⁴ (3²)² = 3²ˣ² = 3⁴ Step 3: Substitute these back into the expression: [ 3¹⁴ × 5³ ] / [ 3⁴ × 5² ]. Step 4: Use the division rule aᵐ ÷ aⁿ = aᵐ⁻ⁿ for the terms with base 3 and base 5 separately. For base 3: 3¹⁴ ÷ 3⁴ = 3¹⁴⁻⁴ = 3¹⁰. For base 5: 5³ ÷ 5² = 5³⁻² = 5¹. Final answer: 3¹⁰ × 5¹ or 3¹⁰ × 5
  • Q: Find the value of m if (-3)ᵐ⁺¹ × (-3)⁵ = (-3)⁷. A: Step 1: The equation involves multiplication of powers with the same base (-3). Apply the law aᵐ × aⁿ = aᵐ⁺ⁿ to the left side. (-3)⁽ᵐ⁺¹⁾⁺⁵ = (-3)⁷ Step 2: Simplify the exponent on the left side. (-3)ᵐ⁺⁶ = (-3)⁷ Step 3: Since the bases are the same, the exponents must be equal for the equation to be true. So, m + 6 = 7. Step 4: Solve for m by subtracting 6 from both sides. m = 7 - 6 = 1. Final answer: m = 1
  • Q: Simplify: (2⁰ + 3⁰) × (4⁰ + 5⁰) A: Step 1: Use the zero exponent rule, which states that any non-zero number raised to the power of zero is 1 (a⁰ = 1). Apply this to each term: 2⁰ = 1, 3⁰ = 1, 4⁰ = 1, and 5⁰ = 1. Step 2: Substitute these values back into the expression. (1 + 1) × (1 + 1) Step 3: Perform the addition inside the brackets. 2 × 2 Step 4: Perform the final multiplication. 2 × 2 = 4. Final answer: 4

Frequently Asked Questions

What is the difference between an exponent and a power?

The 'exponent' is the small number that indicates how many times the base is multiplied by itself. The 'power' usually refers to the entire expression (base and exponent together), like 5³, or the result of the calculation. They are often used interchangeably.

Why is any number to the power of zero equal to 1?

Think about the division law: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. If m = n, then we have aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰. We also know that any number divided by itself is 1. Therefore, a⁰ must be equal to 1.

How are exponents used in real life?

Exponents are used everywhere! Scientists use them to describe huge distances in space (light-years) or tiny sizes of atoms. Computer scientists use powers of 2 (binary) for data storage (kilobytes, megabytes, gigabytes). They are also used in finance for calculating compound interest.