12th Com Maths Part 2 Chapter 5 (Digest) Maharashtra state board

Chapter 5 Index numbers

Open with Full Screen in HD Quality

HD PDF

Project on Index numbers

Index numbers in mathematics, also known as exponents or powers, represent the number of times a base number is multiplied by itself. They are used to simplify expressions involving repeated multiplication. Here is a detailed explanation of index numbers:

Basic Concept

The notation $a^n$ is used to denote a number $a$ raised to the power of $n$. In this expression:

• $a$ is the base.
• $n$ is the exponent or index.

For example, $2^3$ means $2 \times 2 \times 2 = 8$.

Properties of Exponents

1. Multiplication of Like Bases:

   $$a^m \times a^n = a^{m+n}$$

   Example: $2^3 \times 2^4 = 2^{3+4} = 2^7$.

2. Division of Like Bases:

   $$\frac{a^m}{a^n} = a^{m-n} \quad \text{(for \( a \neq 0 \))}$$

   Example: $\frac{2^5}{2^3} = 2^{5-3} = 2^2 = 4$.

3. Power of a Power:

   $$(a^m)^n = a^{mn}$$

   Example: $(2^3)^2 = 2^{3 \times 2} = 2^6 = 64$.

4. Product of Powers:

   $$(ab)^n = a^n \times b^n$$

   Example: $(2 \times 3)^2 = 2^2 \times 3^2 = 4 \times 9 = 36$.

5. Quotient of Powers:

   $$\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \quad \text{(for \( b \neq 0 \))}$$

   Example: $\left( \frac{2}{3} \right)^2 = \frac{2^2}{3^2} = \frac{4}{9}$.

6. Zero Exponent:

   $$a^0 = 1 \quad \text{(for \( a \neq 0 \))}$$

   Example: $5^0 = 1$.

7. Negative Exponent:

   $$a^{-n} = \frac{1}{a^n} \quad \text{(for \( a \neq 0 \))}$$

   Example: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

8. Fractional Exponent:

   $$a^{\frac{m}{n}} = \sqrt[n]{a^m}$$

   Example: $8^{\frac{1}{3}} = \sqrt[3]{8} = 2$.

Special Cases

• Square Root:

  $$\sqrt{a} = a^{\frac{1}{2}}$$

  Example: $\sqrt{9} = 9^{\frac{1}{2}} = 3$.

• Cube Root:

  $$\sqrt[3]{a} = a^{\frac{1}{3}}$$

  Example: $\sqrt[3]{27} = 27^{\frac{1}{3}} = 3$.

Applications

Index numbers are used in various mathematical and scientific contexts:

• Algebra: Simplifying polynomial expressions, solving exponential equations.
• Calculus: Differentiating and integrating exponential functions.
• Physics: Describing exponential growth/decay processes.
• Engineering: Calculations involving power laws and scaling laws.

Understanding the rules and properties of exponents allows for efficient manipulation and simplification of mathematical expressions.