Introduction - Exponents and Powers - Chapter 10, NCERT Class 8th Maths

Exponents and Powers: Laws, Negative Exponents, and Scientific Notation

An exponent (or power) tells us how many times a number — called the base — is multiplied by itself. For example, 2³ = 2 × 2 × 2 = 8, where 2 is the base and 3 is the exponent. Exponents provide a compact way to express very large and very small numbers, and their laws simplify calculations that would otherwise be extremely tedious. This chapter in CBSE Class 8 Mathematics introduces the fundamental laws of exponents, the concept of negative and zero exponents, and the use of scientific notation to handle measurements that span enormous ranges of magnitude.

The seven fundamental laws of exponents are: (1) aᵐ × aⁿ = aᵐ⁺ⁿ — when multiplying powers with the same base, add the exponents (e.g., 2³ × 2⁴ = 2⁷ = 128). (2) (aᵐ)ⁿ = aᵐⁿ — when raising a power to a power, multiply the exponents (e.g., (3²)⁴ = 3⁸). (3) (ab)ⁿ = aⁿbⁿ — when a product is raised to a power, distribute the exponent (e.g., (2 × 3)² = 2² × 3² = 36). (4) aᵐ/aⁿ = aᵐ⁻ⁿ — when dividing powers with the same base, subtract the exponents (e.g., 5⁵/5³ = 5² = 25). (5) (a/b)ⁿ = aⁿ/bⁿ — a fraction raised to a power distributes the exponent to both numerator and denominator. (6) a⁰ = 1 — any non-zero number raised to the power of zero is 1 (this follows from aᵐ/aᵐ = aᵐ⁻ᵐ = a⁰ = 1). (7) a⁻ⁿ = 1/aⁿ — a negative exponent means "take the reciprocal" (e.g., 2⁻³ = 1/2³ = 1/8).

Negative exponents extend the concept of powers to fractions and reciprocals. They are not negative numbers — the minus sign in the exponent is an instruction to take the reciprocal: a⁻ⁿ = 1/aⁿ. This means 10⁻³ = 0.001, 5⁻² = 1/25 = 0.04, and (3/4)⁻¹ = 4/3. Zero exponents follow logically from the division law: a⁰ = aⁿ/aⁿ = aⁿ⁻ⁿ = a⁰ = 1, provided a ≠ 0. The value of 0⁰ is undefined. Scientific notation (also called standard form) expresses any number as a product of a number between 1 and 10 and a power of 10: a × 10ⁿ, where 1 ≤ a < 10. This is essential for expressing quantities that are very large (the speed of light ≈ 3 × 10⁸ m/s, the population of India ≈ 1.4 × 10⁹) or very small (the mass of an electron ≈ 9.1 × 10⁻³¹ kg, the thickness of a soap bubble ≈ 10⁻⁷ m). Comparing numbers in scientific notation: if n₁ > n₂, then 10ⁿ¹ > 10ⁿ². If the exponents are equal, compare the decimal parts. Writing in standard form requires adjusting the decimal point so that the coefficient lies between 1 and 10, adjusting the exponent accordingly. Board problems involve simplifying expressions with multiple operations on exponents, converting between standard form and expanded form, and comparing very large or very small quantities.

• Fundamental laws: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ/aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; (ab)ⁿ = aⁿbⁿ; (a/b)ⁿ = aⁿ/bⁿ.
• Special cases: a⁰ = 1 (any non-zero number to power 0); a⁻ⁿ = 1/aⁿ (negative exponent means reciprocal).
• Scientific notation: any number = a × 10ⁿ where 1 ≤ a < 10 — essential for expressing very large and very small quantities.
• Converting to standard form: adjust decimal point so coefficient is between 1 and 10, modify exponent to compensate.
• Comparing numbers in scientific notation: first compare exponents; if equal, compare the decimal coefficients.