Syllabus Numbers & Quantification Indices and its properties

Indices and its properties

A comprehensive guide on Indices and their laws for CBSE Class 11 Applied Mathematics, covering exponents, powers, and simplification techniques with detailed examples.

Fundamental Laws of Indices

Indices, also known as exponents or powers, represent the number of times a base is multiplied by itself. Understanding the properties of indices is essential for simplifying algebraic expressions and solving exponential equations.

1. Product Rule: $a^m \times a^n = a^{m+n}$ | 2. Quotient Rule: $a^m / a^n = a^{m-n}$ | 3. Power of a Power: $(a^m)^n = a^{m \times n}$ | 4. Zero Exponent: $a^0 = 1$ (a ≠ 0) | 5. Negative Exponent: $a^{-n} = 1 / a^n$
Example 1: Simplify the expression: (2^3 × 2^5) / 2^4
Show Step-by-Step Solution

• Apply the product rule to the numerator: 2^3 × 2^5 = 2^(3+5) = 2^8
• Apply the quotient rule to the result: 2^8 / 2^4 = 2^(8-4)
• Calculate the final exponent: 2^4 = 16

Answer: The simplified value is 16.

Fractional Indices and Radicals

When an exponent is a fraction, it represents a root of the base. Specifically, $a^{1/n}$ is equivalent to the nth root of 'a'. These are used to convert radical forms into exponential forms for easier calculation.

a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m
Example 1: Evaluate: 8^(2/3)
Show Step-by-Step Solution

• Express the base 8 as a power: 8 = 2^3
• Substitute into the expression: (2^3)^(2/3)
• Apply the power of a power rule: 2^(3 * 2/3)
• Simplify the exponent: 2^2 = 4

Answer: The value of 8^(2/3) is 4.