Problem 58 - Prealgebra

Question

Multiply the decimal by the given power of 10 . \(65.722 \cdot 100\)

Step-by-Step Solution

Answer

The result is 6572.2.

Step 1: Identify the Decimal Number

The decimal number we need to work with is 65.722.

Step 2: Identify the Power of 10

The power of ten in this problem is 100, which is equivalent to \(10^2\).

Step 3: Determine Decimal Movement

To multiply by \(10^n\), move the decimal point in the number to the right by n places. Here, n is 2, so the decimal point in 65.722 needs to move 2 places to the right.

Step 4: Move the Decimal Point

Starting with 65.722, move the decimal point 2 places to the right. This results in the number 6572.2 after moving the decimal point.

Key Concepts

Powers of 10

Working with powers of 10 is a foundational concept in mathematics that simplifies various operations, particularly multiplication. Powers of 10 refer to numbers like 10, 100, 1000, and so on, which can be mathematically represented as \(10^n\), where \(n\) is a non-negative integer. For instance, \(10^1\) equals 10, \(10^2\) equals 100, and \(10^3\) equals 1000.

Understanding this concept helps in quickly performing operations with large or small numbers. Multiplying by a power of 10 simply involves manipulating the decimal point in the number you are working with, as opposed to performing long arithmetic calculations. When you see a power of 10, you are dealing with zeros that define how many places you will shift the decimal point in other numbers. This approach makes calculations faster, paving the way for handling complex problems with ease.

Decimal Point Movement

Decimal point movement is a straightforward yet critical process when multiplying decimals by powers of 10. This involves shifting the decimal point to the right based on the power exponent of 10 you are multiplying by. Understanding this movement helps in enhancing efficiency and accuracy in calculations that involve non-whole numbers.

Here’s how it works:

• Determine the power of 10. For instance, in \(10^2\), the exponent is 2.
• Move the decimal point in the original number 2 places to the right.
• Add zeros if necessary to fill any gaps after moving the decimal point.

Take 65.722 as an example: when multiplying by 100 (\(10^2\)), the decimal moves two places to the right, converting it to 6572.2. This simple movement ensures that you adjust the number without getting into complicated arithmetic.

Multiplication Strategies

Multiplication strategies for decimals, especially involving powers of 10, focus on leveraging simple operations such as shifting decimal points instead of traditional multiplication. These strategies simplify calculations and minimize errors, particularly for students who are still building their foundational math skills.

Here are a few strategies:

• Identify the power of 10 and its exponent to determine decimal point movement.
• Visually count decimal places to ensure accuracy before and after moving the decimal point.
• Practice with different numbers and powers of 10 to grasp the concept thoroughly.
• Check results by converting decimals into fractions if needed, to verify your understanding of how the number changes.

Applying these strategies to the example of multiplying 65.722 by 100, you move the decimal point two places right, swiftly obtaining the result without engaging in complex calculations.