Exponents and Powers – Complete Guide For Class 7 Math Chapter 11

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The chapter on Exponents and Powers introduces students to a powerful way of representing and simplifying large numbers. This chapter covers the basics of exponents, the laws of exponents, and the application of these concepts in various mathematical problems. By understanding and applying the concepts of exponents and powers, students develop a crucial mathematical tool that simplifies calculations and enhances their problem-solving skills. This foundation prepares them for more advanced topics in mathematics and science, where exponents are used extensively.

Exponents and Powers

Let’s understand the definition of Exponents with these examples

• The number of cells in the human body is about 100,000,000,000,000!
• Our universe contains about 50,000,000,000,000,000,000,000 stars!
• The mean distance of the planet Pluto from the Sun is 5,900,016,038,000 meters!

These numbers are incredibly large and difficult to read, understand, or work with. However, these very large numbers can be easily dealt with using exponents.

Exponents

Exponents help express very large numbers in a shorter notation. For instance, 1000 can be written as 10³, where ’10’ is the base, and ‘3’ is the exponent. Similarly, numbers like 10³ and 10⁵ are the exponential forms of 1000 and 100,000, respectively. 

• x × x = x²  (‘x raised to the power 2’ or ‘x squared’)
• x × x × x = x³  (‘x raised to the power 3’ or ‘x cubed’)
• x × x × x × x = x⁴ (‘x raised to the power 4’)
• x × x × x × x × x = x⁵ (‘x raised to the power 5’)
• x × x × x × x × x × x × x × x = x⁸ (‘x raised to the power 8’)
• x × x × y × y × y = x² y³ (‘x squared y cubed’)

Base 1 and Base (-1)

For any whole number 1ᵃ = 1

Thus, 1 raised to the power of any whole number is 1.

• For any odd whole number  (-1)ᵃ = -1
• For any even whole number (-1)ᵃ = 1

Examples: 

• (-1/2)³ = -1 x -1 x -1/ 2 x 2 x 2 = -1/8
• (5/7)³ = 5 x 5 x 5 / 7 x 7 x 7 = 125 / 343

Laws of Exponents

In the chapter Exponents and Powers, there is a mention of various laws of exponents. Those Include:

1. Multiplying Powers with the Same Base:

For any non-zero integer a, where x and y are whole numbers,

𝑎ˣ × 𝑎ʸ = 𝑎ˣ⁺ʸ

Thus, when multiplying powers with the same bases, we simply add the exponents & the base remains as it is, in the product.

Examples:

• 7² × 7¹ = 7²⁺¹ = 7³
• 5³ × 5⁴  = 5³⁺⁴ = 5⁷

2. Dividing Powers with the Same Base:

For any non-zero integer a, where x and y are whole numbers,

𝑎ˣ ÷ 𝑎ʸ = 𝑎ˣ⁻ʸ

Thus, when dividing powers with the same bases, we simply subtract the exponents & the base remains as it is, in the quotient. 

Examples:

•  7² ÷ 7¹ = 7²⁻¹ = 7
•  5⁴ ÷ 5³ = 5⁴⁻³ = 5

3. Taking Power of a Power:

For any non-zero integer a, where x and y are whole numbers,

(𝑎ˣ)ʸ = 𝑎ˣ ˣ ʸ

Thus, while taking the power of a power, we simply multiply the exponents & the base remains as it is.

Example: (4²)³ = 4²ˣ³ = 4⁶

4. Multiplying Powers with the Same Exponents:

For any non-zero integers a and b, where x is a whole number,

 𝑎ˣ × 𝑏ˣ =  (𝑎 × 𝑏)ˣ 

Thus, when multiplying powers with the same exponents, we simply multiply the bases & the exponent remains as it is, in the product. 

Example: 5² x 3²  = (5 x 3)² = 15²

5. Dividing Powers with the Same Exponents:

For any non-zero integers a and b, where x is a whole number,

𝑎ˣ ÷ 𝑏ˣ = (𝑎 ÷ 𝑏)ˣ

Thus, when dividing powers with the same exponents, we simply divide the bases & the exponent remains as it is, in the quotient.

Example: 6⁴ ÷ 3⁴ = (6 ÷ 3)⁴ = 2⁴

6. Numbers with Zero Exponent

For any non-zero integer a,

𝑎⁰ = 1

Thus, any non-zero integer raised to the power 0 is 1. 

Examples:

•  25⁰ = 1
• (8)⁰ = 1
• 𝑧⁰ = 1

Now let’s understand the Decimal Number System

We can express the expansion of any number using powers of 10 in exponent form.

• 87235 = 8 × 10⁴ + 7× 10³ + 2 × 10² + 3 × 10¹ + 5 × 10⁰
• 658100 = 6 × 10⁵ + 5 × 10⁴ + 8 × 10³ + 1 × 10² + 0 × 10¹ + 0 × 10⁰ 

Expressing Large Numbers in Standard Form

The standard form is used by mathematicians and scientists to write very large numbers simply and understandably.

Example:

56423 becomes 5.6423 × 10⁴ 

Thus, 56423 in standard form is 5.6423 × 10⁴ 

Incorrect standard forms for 56423:

• 56.423 × 10³ 
• 564.23 × 10²
• 5642.3 × 10¹ 

In summary, Class 7 Math Chapter 11 – Exponents and Powers provides a foundational understanding of how to handle large numbers and simplify mathematical expressions using exponents. By mastering the principles outlined in this chapter, such as the laws of exponents and their applications, students can significantly enhance their mathematical problem-solving skills.

The chapter on Exponents and Powers introduces key concepts that are essential not only for Class 7 mathematics but also for advanced topics in higher grades. With the help of iPrep’s engaging animated videos, practice exercises, and comprehensive notes, students can thoroughly grasp the chapter’s concepts and apply them with confidence.

Remember, Exponents and Powers are more than just mathematical tools; they are powerful techniques that simplify complex calculations and help in understanding real-world phenomena. Embracing these concepts early on will prepare you for future challenges and contribute to a solid mathematical foundation.

Explore more resources on Exponents and Powers with iPrep to ensure a complete and thorough understanding of this essential topic.