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Chapter 10: Exponents and Powers – CBSE Class 8 Mathematics Detailed Notes

Chapter 10: Exponents and Powers

Introduction to Exponents

Exponents represent the repeated multiplication of the same number. If a is a number and n is a positive integer, then:

an = a × a × … × a (n times)

Example: 23 = 2 × 2 × 2 = 8

Use of Exponents to Express Small Numbers in Standard Form

Very small numbers can also be written in Standard Form using negative exponents. This is particularly useful in scientific notation when dealing with measurements that involve extremely small values.

m × 10−n where 1 ≤ m < 10 and n is a positive integer.

Examples of Expressing Small Numbers Using Exponents in Standard Form

  • 0.000012 is written as 1.2 × 10−5
  • 0.000056 is written as 5.6 × 10−5
  • 0.0045 is written as 4.5 × 10−3
  • 0.0000008 is written as 8 × 10−7
  • 0.00321 is written as 3.21 × 10−3
  • 0.00009 is written as 9 × 10−5
  • 0.00000052 is written as 5.2 × 10−7
  • 0.0000000035 is written as 3.5 × 10−9
  • 0.00502 is written as 5.02 × 10−3
  • 0.00076 is written as 7.6 × 10−4

Comparing Very Large and Very Small Numbers

When comparing numbers that vary greatly in size, it is easier to express them in Standard Form using exponents. The size of the number can be directly compared by looking at the power of 10.

Examples of Comparing Large and Small Numbers

  • 108 = 100,000,000 and 103 = 1,000
  • 10−3 = 0.001 and 10−6 = 0.000001
  • 6.5 × 107 = 65,000,000 and 3.2 × 105 = 320,000
  • 2.5 × 10−4 = 0.00025 and 1.8 × 10−6 = 0.0000018
  • 1.23 × 1012 = 1,230,000,000,000 and 7.8 × 1011 = 780,000,000,000

Laws of Exponents

There are several important laws that simplify operations involving exponents:

Law 1: Multiplying Powers with the Same Base

am × an = am+n

Example: 23 × 24 = 27 = 128

Law 2: Dividing Powers with the Same Base

am ÷ an = am-n

Example: 56 ÷ 52 = 54 = 625

Law 3: Power of a Power

(am)n = am×n

Example: (23)2 = 26 = 64

Law 4: Multiplying Powers with the Same Exponent

an × bn = (a × b)n

Example: 32 × 52 = (3 × 5)2 = 152 = 225

Law 5: Dividing Powers with the Same Exponent

an ÷ bn = (a ÷ b)n

Example: 83 ÷ 43 = (8 ÷ 4)3 = 23 = 8

Negative Exponents

A negative exponent represents the reciprocal of the base raised to the opposite positive exponent.

a−n = 1/an

Example: 3−2 = 1/32 = 1/9

Standard Form

Large Numbers in Standard Form

When dealing with large numbers, it is useful to write them in standard form using powers of 10:

Standard Form = m × 10n where 1 ≤ m < 10

Example: 5,600,000 = 5.6 × 106

Small Numbers in Standard Form

Small numbers (less than 1) can also be expressed in standard form using negative powers of 10:

Example: 0.000045 = 4.5 × 10−5

Expressing Large Numbers Using Exponents

Large numbers are often written more simply using exponents:

  • Example: 10,000 = 104
  • Example: The distance between the Earth and the Sun is approximately 1.496 × 1011 meters.

Comparison of Large Numbers Using Exponents

To compare large numbers expressed with exponents, we compare their powers of 10.

  • Example: Compare 106 and 108. Since 108 has a larger exponent, it is the bigger number.

Scenario-Based Mathematical Problems

Scenario 1: Population Growth

Problem: The population of a town is 105 and it increases by 5% every year. What will be the population after 3 years?

Solution: Use the formula for exponential growth: Population = P × (1 + r)t. The final population is approximately 105 × (1 + 0.05)3.

Scenario 2: Speed of Light

Problem: The speed of light is approximately 3 × 108 meters per second. How long will it take for light to travel from the Sun to the Earth, a distance of 1.496 × 1011 meters?

Solution: Time = Distance ÷ Speed = 1.496 × 1011 ÷ 3 × 108 = 498.67 seconds.

Real-Life Applications of Exponents

  • Astronomy: Distances in space are often expressed in powers of 10.
  • Population Studies: Exponential functions are used to model population growth.
  • Economics: Compound interest problems use exponents to represent the growth of money over time.

Key Terms and Concepts

  • Exponent: The number that indicates how many times the base is multiplied by itself.
  • Base: The number that is multiplied repeatedly in a power.
  • Power: A number expressed using a base and an exponent.
  • Standard Form: A way of writing very large or very small numbers using powers of 10.
  • Reciprocal: The inverse of a number, used when dealing with negative exponents.

Additional Mathematical Problems

Problem 1: Simplifying Exponents

Problem: Simplify 23 × 24 × 2−5.

Solution: 23+4−5 = 22 = 4

Problem 2: Real-Life Scenario – Computer Storage

Problem: A computer’s storage capacity is 210 bytes. How many kilobytes (KB) does the computer have?

Solution: 210 = 1024 bytes. Since 1 KB = 1024 bytes, the computer has 1 KB of storage.

Problem 3: Solving with Standard Form

Problem: Express 0.000027 in standard form.

Solution: 0.000027 = 2.7 × 10−5

Example 2: Converting Small Numbers to Standard Form

Convert 0.000056 into standard form.

Solution: 0.000056 = 5.6 × 10−5

Chapter 10: Exponents and Powers – FAQs

1. What is an exponent?
An exponent refers to the number of times a number (base) is multiplied by itself. For example, in 23, the number 2 is the base and the exponent is 3, which means 2 is multiplied by itself 3 times (2 × 2 × 2).
2. How do you express large numbers using exponents?
Large numbers can be expressed using exponents by writing them in the form of powers of 10. For example, 10,000 can be written as 104, where the exponent 4 represents the number of zeros.
3. How do you express small numbers using exponents?
Small numbers can be expressed using negative exponents. For example, 0.0001 can be written as 1 × 10-4, where the negative exponent indicates the decimal shift.
4. What does it mean if an exponent is negative?
A negative exponent represents the reciprocal of the base raised to the opposite positive exponent. For example, 2-3 is equivalent to 1/23, or 1/8.
5. What is the value of any number raised to the power of 0?
Any number raised to the power of 0 is equal to 1. For example, 50 = 1.
6. What is the product of two powers with the same base?
When multiplying two powers with the same base, the exponents are added. For example, 23 × 24 = 27.
7. How do you divide powers with the same base?
When dividing powers with the same base, the exponents are subtracted. For example, 56 ÷ 52 = 54.
8. What is the power of a power rule?
The power of a power rule states that (am)n = am×n. For example, (23)2 = 26.
9. How do you simplify numbers in standard form?
Numbers in standard form are simplified by expressing them as a number between 1 and 10, multiplied by a power of 10. For example, 5,600,000 can be written as 5.6 × 106.
10. What is 23?
23 = 2 × 2 × 2 = 8.
11. How do you express 0.000045 in standard form?
0.000045 can be expressed as 4.5 × 10-5 in standard form.
12. What is the value of 52?
52 = 25.
13. What is the result of 106 ÷ 102?
106 ÷ 102 = 104.
14. What is the value of (32)3?
(32)3 = 36 = 729.
15. How do you compare very large and very small numbers using exponents?
Large and small numbers can be compared by looking at the powers of 10. For example, 108 is larger than 103, and 10-3 is larger than 10-6.
16. What is the value of 2-2?
2-2 = 1/22 = 1/4.
17. How is the exponent applied when multiplying numbers with the same exponent but different bases?
When multiplying numbers with the same exponent but different bases, the bases are multiplied, and the exponent remains the same. For example, 23 × 53 = (2 × 5)3 = 103.
18. What is the result of (23 ÷ 22)?
(23 ÷ 22) = 21 = 2.
19. What is 10-3 in decimal form?
10-3 = 0.001.
20. What is the standard form of 6,500,000?
The standard form of 6,500,000 is 6.5 × 106.
21. What is 0.00056 expressed in standard form?
0.00056 in standard form is 5.6 × 10-4.
22. What is the reciprocal of 23?
The reciprocal of 23 is 2-3 = 1/8.
23. What is the value of 10-2?
10-2 = 0.01.
24. How do you express 2,500,000 in standard form?
2,500,000 can be expressed as 2.5 × 106.
25. What is 34?
34 = 3 × 3 × 3 × 3 = 81.
MCQs on Chapter 10: Exponents and Powers

MCQs on Chapter 10: Exponents and Powers

1. What is the value of 23?

2. Express 0.000056 in standard form.

3. What is the value of 52?

4. What is the value of 10-2?

5. Which of the following is true about am × an?

6. What is the result of (23)2?

7. Express 0.0045 in standard form.

8. What is the value of 60?

9. Which of the following is true for a0?

10. Which of the following is the reciprocal of 52?

11. What is the value of 2-3?

12. Simplify (53 ÷ 52).

13. What is the value of (3 × 2)2?

14. What is the value of 105 × 103?

15. Simplify 43 ÷ 23.

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