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.

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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 $0.5625$.