Problem 67 - Prealgebra

Question

Multiply the decimal by the given power of 10 . \(74 .896 \cdot 1000\)

Step-by-Step Solution

Answer

The answer is 74896.

Step 1: Understanding the Operation

We are asked to multiply a decimal number, 74.896, by a power of 10, specifically 1000. Multiplying by powers of 10 involves moving the decimal point to the right.

Step 2: Identify the Number of Decimal Places to Move

Since we are multiplying by 1000, which is equal to \(10^3\), we will move the decimal point three places to the right.

Step 3: Move the Decimal Point

Take the number 74.896 and move the decimal point three places to the right. This transforms the number to 74896.0.

Step 4: Write the Final Answer

The original number 74.896, when multiplied by 1000, becomes 74896. The operation does not change the digits beyond moving the decimal.

Key Concepts

Place Value

Place value is a crucial concept in understanding how numbers work, especially when you're dealing with decimals. Every digit in a number has a place, and each place has a value that is ten times the place to its right. This means that in the number 74.896, the '7' is in the tens place, which means it actually represents the value 70, not just 7.

• The '4' is in the ones place and has the value of 4.
• The '8' is in the tenths place and represents 0.8.
• The '9' is in the hundredths place and stands for 0.09.
• The '6' is in the thousandths place, which is equal to 0.006.

Understanding place value helps make sense of the decimal point movement that happens when multiplying by powers of 10. Every move of the decimal point to the right decreases the place value of each digit, which effectively makes the number larger by amplifying the value of each digit.

Powers of 10

The powers of 10 form the backbone of our decimal number system. When we refer to a power of 10, like 1000, we're talking about 10 raised to the power of 3, or written as \(10^3\). This means we are multiplying 10 by itself three times: 10 × 10 × 10. Therefore, 1000 is essentially a way to scale numbers up or down in our base-10 system.

• Each power of 10 corresponds to the number of zeros you would write out: 10 has one zero, 100 has two, and 1000 has three zeros.
• Multiplying by these powers is equivalent to shifting the decimal point a number of places equal to the number of zeros in the power of 10.

The concept makes multiplication straightforward and quick, allowing us to easily grasp huge or tiny numbers merely by adjusting the decimal point.

Decimal Point Movement

Decimal point movement is a straightforward but essential part of multiplying decimals by powers of 10. In our example of multiplying 74.896 by 1000, the decimal point moves three places to the right. This is because 1000 is \(10^3\) and indicates moving the decimal three spots right to scale up the number.

• For each zero in a power of 10, the decimal moves one place.
• Moving to the right amplifies the number, making it larger.
• Conversely, if dividing by a power of 10, the decimal moves left, which reduces the number.

After moving the decimal, the original number, 74.896, transforms to 74896.0. The digits haven't changed; they are merely in different positions of higher value. This method makes handling numbers quick without manually multiplying each digit, simplifying calculations significantly.