Class 8 Notes - Exponents and Powers

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Exponents and Powers

Exponents and powers are a way to express repeated multiplication of the same number. They help us write large and small numbers in a compact form and make calculations easier.

What is an Exponent?

An exponent refers to the number of times a number (called the base) is multiplied by itself.
For example, in \(2^3\), 2 is the base and 3 is the exponent. \(2^3 = 2 \times 2 \times 2 = 8\).

Important Terms

  • Base: The number that is being multiplied.
  • Exponent (or Power): The number of times the base is multiplied by itself.
  • Standard Form: Writing numbers using exponents, e.g., \(10^5\).
  • Expanded Form: Writing the multiplication, e.g., \(10^5 = 10 \times 10 \times 10 \times 10 \times 10\).

Laws of Exponents

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

Standard Form and Scientific Notation

Very large or very small numbers can be written using exponents, especially powers of 10.
Example: 1,000,000 = \(10^6\); 0.0001 = \(10^{-4}\)

Examples

  • \(3^4 = 3 \times 3 \times 3 \times 3 = 81\)
  • \(5^0 = 1\)
  • \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\)
  • \((2^3)^2 = 2^{3 \times 2} = 2^6 = 64\)
  • \(10^5 \times 10^2 = 10^{5+2} = 10^7\)

Word Problems

  1. Express 729 as a power of 3.
    Solution: \(729 = 3^6\)
  2. Write 1/81 as a power of 3.
    Solution: \(1/81 = 3^{-4}\)
  3. Simplify: \(2^5 \times 2^3\).
    Solution: \(2^{5+3} = 2^8 = 256\)
  4. Express 0.001 in standard form.
    Solution: \(0.001 = 1 \times 10^{-3}\)

Practice Exercises

  1. Write \(4 \times 4 \times 4 \times 4\) using exponents.
  2. Express 1/125 as a power of 5.
  3. Simplify: \(7^3 \div 7^2\).
  4. Write 100,000 in exponential form.
  5. Simplify: \((3^2)^4\).

Summary

  • Exponents are used to express repeated multiplication.
  • Laws of exponents help in simplifying expressions.
  • Negative and zero exponents have special meanings.
  • Standard form is useful for writing very large or small numbers.