Problem 57 - Prealgebra

Question

In Exercises 57-68, multiply the decimal by the given power of 10 . \(24.264 \cdot 10\)

Step-by-Step Solution

Answer

242.64

Step 1: Understand the Problem

We need to multiply the decimal number, 24.264, by 10. This involves shifting the decimal point to the right.

Step 2: Shift the Decimal Point

When multiplying a decimal by 10, move the decimal point one place to the right. So, 24.264 becomes 242.64 after shifting the decimal point.

Key Concepts

Decimal Point Shift

When we multiply a decimal number by a power of 10, a simple and efficient method is to shift the decimal point. This avoids performing complex multiplication operations. The number of spaces you move the decimal point depends on the specific power of 10 by which you are multiplying.

• Multiplying by 10 shifts the decimal one place to the right.
• Multiplying by 100 moves it two places to the right.
• Multiplying by 1,000 shifts it three places, and so on.

Essentially, each 0 in the power of 10 provides a guide for how many digits you need to shift. In the case of multiplying 24.264 by 10, the decimal point moves from between the 24 and 264 to after the 4, resulting in 242.64. Shifting the decimal instead of multiplying each digit makes the process quicker and more transparent.

Powers of 10 Multiplication

Powers of 10 play a crucial role in simplifying multiplication with decimals. Multiplying by a power of 10 is akin to scaling the number by tens, hundreds, or thousands, based on the exponent. Whenever you multiply by 10, you are essentially adding one "zero" to the number's value in place of shifting everything right. For instance:

• For 10: Shift once, matching the power of 1 (10^1).
• For 100: Shift twice, due to 10^2.
• This pattern continues for higher powers such as 1,000 (10^3).

Therefore, understanding powers of 10 helps you determine how many places to move the decimal. This understanding provides clarity and efficiency in handling larger and more complex decimal multiplications.

Basic Arithmetic Operations

When performing arithmetic operations with decimals, such as multiplication, division, addition, or subtraction, each operation has rules regarding decimal placement. In multiplication, decimal handling is simplified by using powers of 10. Let's break it down:

• Addition and Subtraction: Line up the decimals and perform the operation as if they are whole numbers.
• Multiplication: Use decimal point shifting for quick calculations, especially with powers of 10.
• Division: Involves adjusting both dividend and divisor to have whole numbers, or shift decimal points to simplify.

Focused simplicity in operations keeps calculations easy to manage. Multiplying with powers of 10 is an excellent example of how basic arithmetic rules help manage and simplify what might seem complex at first. Embracing arithmetic efficiencies paves the way for skillful and confident problem solving.