Exponents & Powers — Class 8

Concise notes explaining exponents (indices), laws of exponents, zero & negative powers, scientific notation, worked examples and practice — mobile-friendly & SEO-optimized.

1. Introduction

An exponent (or index) tells how many times a number (the base) is multiplied by itself. We write a^n meaning a × a × ... × a (n times). Example: 2^3 = 2×2×2 = 8.

2. Important Laws of Exponents

1. a^m × a^n = a^{m+n}
2. \frac{a^m}{a^n} = a^{m−n} (a ≠ 0)
3. (a^m)^n = a^{mn}
4. (ab)^n = a^n b^n
5. \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}
6. (−a)^n = (−1)^n a^n (sign depends on n)
Example: Simplify 2^3 × 2^4 = 2^{3+4} = 2^7 = 128.

3. Zero & Negative Exponents

  • Zero exponent: a^0 = 1 for any a ≠ 0. Example: 5^0 = 1.
  • Negative exponent: a^{−n} = 1/a^n. Example: 2^{−3} = 1/8.
  • Fractional exponents (intro): a^{1/n} = n√a; a^{m/n} = (n√a)^m.

4. Scientific Notation

Used to express very large or very small numbers. Write numbers as m × 10^n where 1 ≤ m < 10 and n is integer. Example: 4500 = 4.5 × 10^3, 0.0062 = 6.2 × 10^{−3}.

Helps in calculating with large/small quantities and is common in science and engineering.

5. Solved Examples

Example 1: Simplify \frac{3^5}{3^2} × 3^{−1}.
Use laws: = 3^{5−2} × 3^{−1} = 3^{2} × 3^{−1} = 3^{1} = 3.
Example 2: Express 0.00042 in scientific notation.
0.00042 = 4.2 × 10^{−4}.
Example 3: Evaluate (−2)^4 and −2^4 (note the difference).
(−2)^4 = 16; −2^4 = −(2^4) = −16 (exponent applies to base only).

6. Practice Questions (with answers)

  1. Simplify: 5^3 × 5^{−2}.
    Answer
    = 5^{3−2} = 5^1 = 5
  2. Write 72000 in scientific notation.
    Answer
    = 7.2 × 10^4
  3. Evaluate: \left(\frac{2}{3}\right)^{−2}.
    Answer
    = \left(\frac{3}{2}\right)^2 = 9/4
  4. Simplify: (x^2 y^3)^2.
    Answer
    = x^{4} y^{6}

Practice using exponent laws step-by-step and watch parentheses and signs carefully — they change the result.

7. Exam Tips & Tricks

  • Apply laws in correct order: simplify exponents before numeric multiplication when possible.
  • Remember a^0 = 1 and a^{−n} = 1/a^n.
  • Be careful with negative signs and parentheses.
  • Use scientific notation to simplify large/small number calculations.