Indices, also known as powers, are a compact way to express repeated multiplication of a number by itself. This section delves into interpreting and manipulating expressions with positive, zero, and negative indices, essential for algebraic competence.

**Introduction to Indices**

The notion of indices (or powers) simplifies the expression of multiplication operations. For instance, $5^3$ signifies $5 \times 5 \times 5$, a concise representation of repeated multiplication.

Image courtesy of APlusTopper

**Positive Indices**

Positive indices denote multiplication repeated according to the power's value:

- $2^4 = 2 \times 2 \times 2 \times 2 = 16$
- $3^3 = 3 \times 3 \times 3 = 27$

**Zero Indices**

A number (except zero) raised to the power of zero equals one, due to the division rule of indices:

- $5^0 = 1$
- $100^0 = 1$

**Negative Indices**

Negative indices indicate the reciprocal of the base raised to the absolute value of the index:

- $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$
- $5^{-1} = \frac{1}{5}$

**Worked Examples**

**Example 1: Simplifying with Positive Indices**

Simplify: $4^3 \times 4^2$.

**Solution:**

$4^3 \times 4^2 = 4^{3+2} = 4^5 = 1024$**Example 2: Working with Zero Indices**

Evaluate: $7^0$.

**Solution:**

$7^0 = 1$**Example 3: Simplifying Expressions with Negative Indices**

Simplify: $3^{-2}$.

**Solution:**

**Practice Questions**

**1. Simplify:** $2^3 \times 2^{-1}$.

#### Solution:

$2^3 \times 2^{-1} = 2^{3-1} = 2^2 = 4$**2. Evaluate:** $10^0$.

#### Solution:

$10^0 = 1$**3. Simplify:** $5^{-3} \times 5^2$.

#### Solution:

$5^{-3} \times 5^2 = 5^{-3+2} = 5^{-1} = \frac{1}{5} = 0.2$**Key Takeaways**

- Positive indices indicate repeated multiplication.
- A non-zero number raised to the power of zero equals one.
- Negative indices represent the reciprocal of the base raised to a positive index.
- Multiplication of powers with the same base involves adding their indices.
- The division of powers with the same base involves subtracting their indices.