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Multiplication is one of the basic arithmetic operations where we combine equal groups to find the total number of items. When we talk about multiplication in the context of exponents, we essentially refer to repeated multiplication of the same number. For example, in the expression \(0.4^{3}\), we are multiplying 0.4 by itself three times. Let's break it down:

• First, multiply 0.4 by 0.4 which equals 0.16.
• Then, take the result (0.16) and multiply it again by 0.4 to get 0.064.

This is a simple application of multiplication involving more than one step.

Exponents are a mathematical notation indicating the number of times a number (the base) is multiplied by itself. In the expression \(0.4^{3}\), '3' is the exponent, and '0.4' is the base. Exponents provide a shorthand for writing repeated multiplication. So, \(0.4^{3}\) means 0.4 multiplied by itself three times. A few key points to remember:

• A positive exponent indicates how many times to multiply the base number.
• An exponent of zero always results in one, regardless of the base (except the base can't be zero).
• Negative exponents indicate reciprocal multiplication.

For the provided exercise, the positive exponent '3' simply tells us to multiply the base, 0.4, by itself three times.

Step-by-step solutions are crucial for understanding mathematical problems thoroughly. For the expression \(0.4^{3}\), the problem was broken down into manageable steps:

1. **Understand the expression**
Recognize that \(0.4^{3}\) means multiplying 0.4 by itself three times.

2. **Perform the multiplication**
Calculate \(0.4 \times 0.4 \times 0.4\).

3. **Multiply the first two terms**
Start with \(0.4 \times 0.4\), which equals 0.16.

4. **Multiply the result by the remaining term**
Next, multiply the result (0.16) by 0.4, which gives 0.064.

5. **Write the final simplified expression**
Therefore, \(0.4^{3} = 0.064\).

Step-by-step solutions help in understanding each action and how they contribute to the final answer. It's like solving a puzzle by piecing together smaller parts in a logical order.