Problem 72 - Prealgebra

Question

Divide the decimal by the given power of 10 . \(\frac{578.68}{100}\)

Step-by-Step Solution

Answer

5.7868

Step 1: Understand the Problem

We need to divide the decimal 578.68 by 100. Dividing by powers of 10, like 10, 100, 1000, etc., involves moving the decimal point to the left.

Step 2: Identify the Power of 10

In this problem, we are dividing by 100. Since 100 is represented as \(10^2\), we will need to move the decimal point 2 places to the left.

Step 3: Move the Decimal Point

Starting with the number 578.68, move the decimal point 2 places to the left. The decimal point is currently after the digit 8 in 578.68. Moving it 2 places to the left, we get 5.7868.

Step 4: Write the Result

After moving the decimal point, the resulting value we obtain is 5.7868. So, \(\frac{578.68}{100} = 5.7868\).

Key Concepts

Powers of Ten

When we talk about powers of ten, we refer to numbers like 10, 100, 1000, and so forth. These numbers are expressed in exponential form as follows:

• 10 is represented as \(10^1\).
• 100 is \(10^2\).
• 1000 is \(10^3\).

Each of these expressions tells us how many times we multiply 10 by itself. For example, \(10^2 = 10 \times 10 = 100\), and \(10^3 = 10 \times 10 \times 10 = 1000\).
These powers determine how many places we move the decimal point when dividing or multiplying, which makes calculations with decimals much simpler and quicker. Rather than performing long division, understanding powers of ten allows us to use quick mental math by simply shifting decimal points.

Decimal Point Movement

Decimal point movement is crucial in simplifying multiplication and division by powers of ten. The direction and the number of places you move the decimal point depend on whether you are multiplying or dividing.

• When dividing by 10, move the decimal point 1 place to the left.
• When dividing by 100, move it 2 places to the left.
• When dividing by 1000, move it 3 places to the left.

For example, let's handle \(578.68 \div 100\). The decimal point starts between the 8 and the 6 in 578.68. Since we are dividing by 100 (which is \(10^2\)), we move the decimal point 2 places to the left.
This results in the number 5.7868, simplifying our division process. This technique bypasses lengthy calculations and highlights the elegance of working with powers of ten.

Division by 100

Specifically, when dividing by 100, you utilize knowledge of powers of ten and decimal point movement together. Because 100 is expressed as \(10^2\), you simply move the decimal point in the number two places to the left to complete the division.
This method is both efficient and straightforward. It ensures accuracy and speeds up the process of dividing decimals. For instance, dividing 578.68 by 100 is easily done by adjusting the decimal point location, giving you 5.7868 in no time at all.
Furthermore, this method of moving the decimal point can be extended to any decimal without having to perform additional complex calculations. This allows for a more intuitive math experience and builds confidence in handling similar mathematical operations in the future.