Complete Guide For Class 8th Math Chapter 12: Exponents and Power

Welcome to iPrep, your Learning Super App. Our learning resources for the chapter, Exponents and Power in Mathematics Class 8th chapter 12 are designed to ensure that you grasp this concept with clarity and perfection. Whether you’re studying for an upcoming exam or strengthening your concepts, our engaging animated videos, practice questions and notes offer you the best of integrated learning with interesting explanations and examples.

Chapter 12 in class 8 introduces exponents and powers, covering the basics of exponential notation, laws of exponents, and the application of these concepts in simplifying expressions. Students learn to express large numbers concisely and perform operations involving powers, fostering a deeper understanding of mathematical growth and patterns.

In this chapter, we explore the fascinating world of exponents and powers, a fundamental concept in mathematics that simplifies the representation of large numbers and complex calculations. Let’s dive into the basics, examples, and laws governing exponents.

Introduction to Exponents and Powers

Have you ever wondered if there’s an easier way to remember and express large numbers? For instance, the mass of the Earth is approximately 5,980,000,000,000,000,000,000 kilograms. That’s a lot to write and remember! Thankfully, there’s a simpler way: using exponents.

The mass of the Earth can be conveniently expressed as 5.98×10²⁴ kg. This expression is read as “5.98 into 10 to the power of 24” or “10 raised to the power of 24.”

Examples of Exponents in Action

Let’s see some basic examples:

• 2² = 2×2 = 4
• 2³ = 2×2×2 = 8
• 2⁴ = 2×2×2×2 = 16
• 2⁵ = 2×2×2×2×2 = 32 

Here, 2 is the base and the number above it (the exponent) tells us how many times to multiply the base by itself.

Powers with Negative Exponents

Can exponents be negative? Yes, they can! For example:

• 5¹ = 5 
• 5² = 5×5 = 25
• 5⁰ = 1 (any number to the power of zero is 1)
• 5⁻¹ = 1/5.  ​Negative exponents indicate reciprocals. Thus, 5⁻¹ means 1/5.

More Examples

Find the value of the following exponents:

1. 7³ = 7×7×7 = 343
2. (−2/3)³ = −2/3 × (−2/3) × (−2/3) = −8/27​
3. (−5)⁻¹ = 1/(−5) = −0.2
4. (3/2)³ =( 3×3×3 )/2×2×2 = 27/8​
5. (−2)⁵ = −2 × (−2) × (−2) × (−2) × (−2) = −32

Laws of Exponents

There are certain universally accepted laws of exponents that simplify calculations:

Law 1: Product of Powers

For the same base, add the exponents:

aᵐ × aⁿ = aᵐ ⁺ ⁿ 

For example, 2³ × 2⁵ = 2 ³⁺⁵ = 2⁸

Law 2: Quotient of Powers

For the same base, subtract the exponents:

aᵐ / aⁿ = aᵐ ⁻ ⁿ 

For example, 2⁸ / 2³ = 2 ⁸⁻³ = 2⁶   

Law 3: Power of a Power

Multiply the exponents:

(aᵐ)ⁿ = a ᵐ ˣ ⁿ

For example, (2³)² = 2³ ˣ ² = 2⁶

Law 4: Powers of Products

For different bases with the same exponent:

(ab)ᵐ = aᵐ × bᵐ

For example, 2³ × 3³ = (2 x 3)³ = 6³.

Law 5: Powers of Fractions

For different bases in fraction form with the same exponent:

(a/b)ᵐ = aᵐ/bᵐ 

For example, (3/5)³ = 3³/5³ = 27/125

Zero Exponent Rule

Any non-zero number raised to the power of zero equals one:

a⁰ = 1

Standard Form for Large and Small Numbers

Any large or small number can be conveniently expressed in standard form using exponents. For example, the number 0.000035 can be written as 3.5×10⁻⁵.

To express numbers in standard form:

1. Move the decimal point so that the number is between 1 and 10.
2. Count the number of places the decimal point has moved; this becomes the exponent.
3. If the decimal point moves to the right, the exponent is negative; if it moves to the left, the exponent is positive.

Examples:

• 0.000000000000009580.000000000000009580.00000000000000958 becomes 9.58×10⁻¹⁵
• 314600000000000003146000000000000031460000000000000 becomes 3.146×10¹⁶

These laws and examples provide a solid foundation for understanding and applying exponents and powers. Mastery of these concepts will make handling large numbers and complex calculations much simpler and more intuitive.