Problem 18 - Algebra A Combined Function

Question

Evaluate. $$ (0.4)^{3} $$

Step-by-Step Solution

Answer

(0.4)^3 = 0.064

Step 1: Understand the Problem

We are asked to evaluate the expression \((0.4)^{3}\). This means we need to calculate the cube of 0.4.

Step 2: Multiply the Number by Itself Three Times

To find \((0.4)^{3}\), we multiply 0.4 by itself two more times: \(0.4 \times 0.4 \times 0.4\).

Step 3: Calculate First Multiplication

First, calculate the result of multiplying 0.4 by 0.4: \(0.4 \times 0.4 = 0.16\).

Step 4: Calculate Second Multiplication

Next, multiply the result from Step 3 by 0.4: \(0.16 \times 0.4 = 0.064\).

Step 5: Verify the Calculation

Double-check the calculations to ensure accuracy. The two multiplications should be correct: \(0.4 \times 0.4 = 0.16\) and \(0.16 \times 0.4 = 0.064\). Thus, \((0.4)^{3} = 0.064\).

Key Concepts

Multiplication

Understanding multiplication is key to finding the cube of a number. Multiplication is the process of finding the total of one number added to itself a certain number of times. When you multiply numbers together, you are essentially adding that number repeatedly.
For example, if you need to multiply 0.4 by itself twice as done in the expression \((0.4)^3\), you are using multiplication to find the result. Here’s a simple breakdown:

• Start with the number: 0.4
• Multiply it by itself: \(0.4 \times 0.4 = 0.16\)
• Finally, multiply the result by the number again: \(0.16 \times 0.4 = 0.064\)

This method ensures that you accurately calculate powers, like cubes, through repeated multiplication.

Exponents

Exponents are a shorthand way to show how many times a number, known as the base, is multiplied by itself. The expression \(0.4^3\) uses an exponent to indicate that 0.4 is multiplied by itself three times.
In the context of our example, the exponent is 3, telling us to cube the number 0.4, or perform the operation \(0.4 \times 0.4 \times 0.4\). Exponents are a powerful mathematical tool because they allow you to perform multiple multiplications quickly and efficiently. Understanding exponents can simplify complex multiplication tasks, and they are especially handy when working with large numbers or fractions.

Decimal Numbers

Decimals are a way of representing numbers that are not whole, using a dot or point to separate the whole number from the fractional part. When dealing with operations involving decimal numbers, it’s crucial to ensure precision in your calculations, especially with multiplication.
In our example, we work with the decimal number 0.4. To find the cube of this decimal, accurate multiplication steps ensure the correct placement of the decimal point in the result. Steps such as multiplying \(0.4 \times 0.4 = 0.16\), and then \(0.16 \times 0.4 = 0.064\) illustrate how to carefully handle decimals while maintaining accuracy. Being meticulous with decimals is crucial as they often appear in real-world applications involving money, measurements, and scientific data.