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.

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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:

Example 2: Working with Zero Indices

Evaluate: $7^0$.

Solution:

Example 3: Simplifying Expressions with Negative Indices

Simplify: $3^{-2}$.

Solution:

Practice Questions

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

Solution:

2. Evaluate: $10^0$.

Solution:

3. Simplify: $5^{-3} \times 5^2$.

Solution:

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.