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Mathematics Class 8 - Exponents And-Powers Notes

Comprehensive study notes for Class 8 - Exponents And-Powers olympiad preparation

Exponents and Powers

Welcome to the chapter on Exponents and Powers for Class 8. In this chapter, you will learn what exponents and powers are, how to use their laws, and how to solve problems using them. By the end of this chapter, you will be able to simplify expressions and solve sums confidently!

Introduction

Exponents and powers are used to write large numbers in a short way. They help us multiply the same number many times easily.

What are Exponents?

An exponent tells us how many times to multiply a number by itself. For example, \(2^3 = 2 \times 2 \times 2 = 8\).

  • Base: The number being multiplied (e.g., 2).
  • Exponent: The number of times the base is multiplied (e.g., 3).

Laws of Exponents

  • Product Law: \(a^m \times a^n = a^{m+n}\)
  • Quotient Law: \(a^m \div a^n = a^{m-n}\)
  • Power Law: \((a^m)^n = a^{m \times n}\)
  • Zero Exponent: \(a^0 = 1\) (where \(a \neq 0\))
  • Negative Exponent: \(a^{-n} = \frac{1}{a^n}\)

Standard Form

Very large or very small numbers can be written using exponents, called standard form or scientific notation.
Example: \(3,000 = 3 \times 10^3\)

Simplifying Expressions

Use the laws of exponents to simplify expressions.

  • Example: \(2^4 \times 2^3 = 2^{4+3} = 2^7 = 128\)
  • Example: \(5^6 \div 5^2 = 5^{6-2} = 5^4 = 625\)

Fun Activity: Exponent Challenge!

Write the number 1,000,000 using exponents. Try to simplify expressions with your friends and see who gets the answer first!

Summary

  • Exponents show repeated multiplication.
  • Use laws of exponents to simplify expressions.
  • Standard form helps write big and small numbers easily.

Practice Questions

  1. Simplify: \(3^2 \times 3^4\)
  2. Write \(2^5\) in expanded form.
  3. Simplify: \(7^6 \div 7^2\)
  4. What is the value of \(10^0\)?
  5. Express 10,000 in standard form using exponents.

Challenge Yourself

  • Simplify: \((4^3)^2\)
  • Write 0.001 in standard form using exponents.

Did You Know?

  • Exponents are used in computers, science, and engineering to handle big numbers!
  • Negative exponents show division by the base.

Glossary

  • Exponent: The number that tells how many times to multiply the base.
  • Base: The number being multiplied.
  • Standard Form: Writing numbers using powers of 10.

Answers to Practice Questions

  1. \(3^{2+4} = 3^6 = 729\)
  2. \(2 \times 2 \times 2 \times 2 \times 2 = 32\)
  3. \(7^{6-2} = 7^4 = 2401\)
  4. 1
  5. \(1 \times 10^4\)

Practice exponents and powers to make math easy and fun!

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