Indices, or powers, are a fundamental concept in mathematics that allows us to express repeated multiplication of a number by itself in a concise manner. This section delves into understanding and using indices, particularly focusing on positive, zero, and negative integers, which are pivotal for solving a variety of mathematical problems efficiently.

Image courtesy of Argoprep

**What are Indices?**

Indices (singular: index), also known as exponents or powers, provide a way to represent the repeated multiplication of a number. For example, $5^3$ (read as 'five cubed') means $5 \times 5 \times 5$.

**Positive Indices**represent standard multiplication, e.g., $2^3 = 2 \times 2 \times 2$.**Zero Index**means any non-zero number raised to the power of zero is 1, e.g., $4^0 = 1$.**Negative Indices**represent the reciprocal of the base raised to the positive power, e.g., $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

**Basic Rules of Indices**

Understanding the rules of indices is crucial for simplifying expressions and solving equations efficiently.

**1. Multiplication Rule**: When multiplying two powers with the same base, add their exponents.

**2. Division Rule**: When dividing two powers with the same base, subtract the exponents.

**3. Power of a Power Rule**: To raise a power to another power, multiply the exponents.

**4. Zero Power Rule**: Any non-zero number raised to the power of zero is 1.

**5. Negative Power Rule**: A negative exponent indicates the reciprocal of the base raised to the positive exponent.

**Application in Practical Situations**

Applying indices in practical situations involves understanding and manipulating various forms of numbers, including negative numbers, improper fractions, mixed numbers, and considering changes such as in temperature.

**Negative Numbers**

**Understanding Negative Indices**: Negative indices can be used to represent division or fractions. For instance, $3^{-2} = \frac{1}{3^2} = \frac{1}{9}$.

**Improper Fractions and Mixed Numbers**

**Converting and Calculating**: Indices can simplify the process of working with improper fractions and mixed numbers. For example, $(\frac{3}{2})^2 = \frac{3^2}{2^2} = \frac{9}{4}$.

**Temperature Changes**

**Calculating Temperature Changes**: Indices may not directly apply to temperature changes but understanding operations with negative numbers can help interpret temperature decreases or increases in certain contexts.

## Worked Examples

**Example 1: Basic Indices Calculation**

Calculate $2^3 \times 2^{-2}$.

**Solution**:

1. Apply the multiplication rule: $2^3 \times 2^{-2} = 2^{3 + (-2)}$

2. Simplify the exponent: $2^{3 + (-2)} = 2^1$

3. Calculate the power: $2^1 = 2$

The answer is $2$, as confirmed by the mathematical calculation.

**Example 2: Working with Fractions**

Simplify $(\frac{4}{3})^{-2}$.

**Solution**:

1. Apply the negative exponent rule: $(\frac{4}{3})^{-2} = (\frac{3}{4})^2$

2. Calculate the square: $(\frac{3}{4})^2 = \frac{3^2}{4^2} = \frac{9}{16}$

The simplified form is $\frac{9}{16}$, which is approximately $0.5625$.