Class 7 Mathematics Notes Chapter 13 (Exponents and Powers) – Mathematics Book

Alright class, let's get started with a very important chapter for your foundational mathematics, especially relevant for competitive exams: Chapter 13 - Exponents and Powers. Understanding this chapter well makes handling large numbers and algebraic manipulations much easier.

Chapter 13: Exponents and Powers - Detailed Notes

1. Introduction: What are Exponents and Powers?

Imagine you need to write the mass of the Earth. It's approximately 5,970,000,000,000,000,000,000,000 kg. Writing and reading such large numbers is cumbersome. Exponents provide a concise way to represent very large (or very small) numbers and repeated multiplications.

• Repeated Multiplication: When a number is multiplied by itself multiple times, we can express it in a shorter form using exponents.
  • Example: 10 × 10 × 10 × 10 can be written as 10⁴.

2. Core Concepts:

• Base: The number that is being multiplied repeatedly. In 10⁴, the base is 10.
• Exponent (or Index or Power): The number that indicates how many times the base is multiplied by itself. In 10⁴, the exponent is 4.
• Power: The entire expression (like 10⁴) is called the power. It represents the value obtained after the repeated multiplication (10⁴ = 10,000).
• Reading Powers:
  • 10⁴ is read as "10 raised to the power of 4" or "the 4th power of 10".
  • Special cases: 5² is often read as "5 squared", and 7³ is often read as "7 cubed".
• Exponential Form: Writing a number using a base and an exponent (e.g., 10⁴, 2⁵, aᵐ).
• Expanded Form: Writing out the repeated multiplication (e.g., 2⁵ = 2 × 2 × 2 × 2 × 2).

Example:
Express 256 using powers of 2.

• 256 = 2 × 128 = 2 × 2 × 64 = 2 × 2 × 2 × 32 = 2 × 2 × 2 × 2 × 16
• = 2 × 2 × 2 × 2 × 2 × 8 = 2 × 2 × 2 × 2 × 2 × 2 × 4 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
• So, 256 = 2⁸ (Here, base = 2, exponent = 8)

3. Powers with Negative Bases:

• If a negative number is raised to an even exponent, the result is positive.
  • Example: (-2)⁴ = (-2) × (-2) × (-2) × (-2) = 4 × 4 = 16
• If a negative number is raised to an odd exponent, the result is negative.
  • Example: (-2)³ = (-2) × (-2) × (-2) = 4 × (-2) = -8

Important Note: Be careful with parentheses!

• (-2)⁴ = 16
• -2⁴ = -(2 × 2 × 2 × 2) = -16

4. Laws of Exponents:

These laws are fundamental for simplifying expressions involving exponents. Let 'a' and 'b' be non-zero integers (or rational numbers later), and 'm' and 'n' be whole numbers.

• Law 1: Multiplying Powers with the Same Base

  • aᵐ × aⁿ = aᵐ⁺ⁿ
  • Explanation: When multiplying powers with the same base, keep the base and add the exponents.
  • Example: 3² × 3⁵ = 3²⁺⁵ = 3⁷

• Law 2: Dividing Powers with the Same Base

  • aᵐ ÷ aⁿ = aᵐ⁻ⁿ (where m > n and a ≠ 0)
  • Explanation: When dividing powers with the same base, keep the base and subtract the exponents (exponent of the denominator from the exponent of the numerator).
  • Example: 5⁶ ÷ 5² = 5⁶⁻² = 5⁴

• Law 3: Taking Power of a Power

  • (aᵐ)ⁿ = aᵐˣⁿ
  • Explanation: When raising a power to another power, keep the base and multiply the exponents.
  • Example: (7²)³ = 7²ˣ³ = 7⁶

• Law 4: Multiplying Powers with the Same Exponent

  • aᵐ × bᵐ = (a × b)ᵐ
  • Explanation: When multiplying powers with different bases but the same exponent, multiply the bases and keep the exponent.
  • Example: 2⁴ × 3⁴ = (2 × 3)⁴ = 6⁴

• Law 5: Dividing Powers with the Same Exponent

  • aᵐ ÷ bᵐ = (a / b)ᵐ (or (a/b)ᵐ) (where b ≠ 0)
  • Explanation: When dividing powers with different bases but the same exponent, divide the bases and keep the exponent.
  • Example: 8⁵ ÷ 4⁵ = (8/4)⁵ = 2⁵

• Law 6: Number with Exponent Zero

  • a⁰ = 1 (where a ≠ 0)
  • Explanation: Any non-zero number raised to the power of zero is equal to 1. (This follows from Law 2: aᵐ ÷ aᵐ = aᵐ⁻ᵐ = a⁰. Also, aᵐ ÷ aᵐ = 1).
  • Example: 100⁰ = 1, (-5)⁰ = 1

5. Expressing Large Numbers in Standard Form (Scientific Notation):

Standard form is used to write very large numbers conveniently. A number is expressed as a decimal number between 1.0 and 10.0 (including 1.0) multiplied by a power of 10.

• General Form: k × 10ⁿ, where 1 ≤ k < 10 and n is a whole number.

Steps to Convert to Standard Form:

1. Place the decimal point after the first non-zero digit from the left.
2. Count the number of places the decimal point has been moved from its original position (at the end of a whole number) to the new position. This count gives you the exponent (n) for the power of 10.

Example:
Express 3,45,000 in standard form.

1. Place the decimal after the first digit (3): 3.45000
2. The decimal point moved 5 places to the left (from the end after the last zero).
3. So, 3,45,000 = 3.45 × 10⁵

Example:
Express 405,000,000 in standard form.

1. Place decimal after 4: 4.05000000
2. Decimal moved 8 places to the left.
3. So, 405,000,000 = 4.05 × 10⁸

Comparing Numbers in Standard Form:

• First, compare the powers of 10. The number with the higher power of 10 is larger.
• If the powers of 10 are the same, compare the decimal numbers (k). The one with the larger decimal part is larger.
  • Example: Compare 4 × 10¹⁴ and 3 × 10¹⁷. Since 17 > 14, 3 × 10¹⁷ is larger.
  • Example: Compare 2.98 × 10⁸ and 3.1 × 10⁸. Since the powers of 10 are the same, compare 2.98 and 3.1. Since 3.1 > 2.98, 3.1 × 10⁸ is larger.

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on the concepts covered. Choose the correct option.

1. The exponential form of 6 × 6 × 6 × 6 is:
   (A) 4⁶
   (B) 6⁴
   (C) 6⁶
   (D) 4⁴

2. The value of (-3)⁴ is:
   (A) 12
   (B) -12
   (C) 81
   (D) -81

3. Simplify: 2³ × 2⁵
   (A) 2⁸
   (B) 2¹⁵
   (C) 4⁸
   (D) 4¹⁵

4. Simplify: (5²)³
   (A) 5⁵
   (B) 5⁶
   (C) 10³
   (D) 25³

5. Which law of exponents is used in the expression aᵐ × bᵐ = (ab)ᵐ?
   (A) Multiplying powers with the same base
   (B) Dividing powers with the same base
   (C) Taking power of a power
   (D) Multiplying powers with the same exponent

6. The value of 7⁰ + 8⁰ + 9⁰ is:
   (A) 24
   (B) 3
   (C) 1
   (D) 0

7. Simplify: 10⁸ ÷ 10⁵
   (A) 10¹³
   (B) 10⁴⁰
   (C) 1³
   (D) 10³

8. Express 512 as a power of 2.
   (A) 2⁷
   (B) 2⁸
   (C) 2⁹
   (D) 2¹⁰

9. The standard form of the number 76,450,000 is:
   (A) 7.645 × 10⁸
   (B) 7.645 × 10⁷
   (C) 76.45 × 10⁶
   (D) 0.7645 × 10⁸

10. Which of the following is true?
    (A) 2³ > 3²
    (B) 2⁴ = 4²
    (C) 2⁵ < 5²
    (D) 10² > 2¹⁰

Answer Key for MCQs:

1. (B)
2. (C)
3. (A)
4. (B)
5. (D)
6. (B)
7. (D)
8. (C) (2⁹ = 512)
9. (B)
10. (B) (2⁴ = 16, 4² = 16)

Make sure you practice applying these laws extensively. They form the building blocks for many advanced mathematical concepts. Good luck with your preparation!