Problem 68 - Prealgebra

Question

Multiply the decimal by the given power of 10 . \(30.728 \cdot 100\)

Step-by-Step Solution

Answer

3072.8

Step 1: Understand the Problem

We need to multiply the decimal number 30.728 by 100. Understanding how multiplication with powers of 10 works is crucial.

Step 2: Identify Movement of Decimal Point

When multiplying by 100, which is 10 squared, we move the decimal point two places to the right.

Step 3: Move the Decimal Point

Starting at 30.728, move the decimal two places to the right. The decimal moves past the '7' and then the '2', resulting in 3072.8.

Step 4: Write the Final Result

After moving the decimal point, the resulting number is 3072.8.

Key Concepts

Powers of 10

Using \[ 100 = 10^2 \] underscores the significance of powers of 10 in mathematics. When we talk about powers of 10, such as 10 to the power 1 (\(10^1\)), 10 to the power 2 (\(10^2\)), and so forth, we're referring to repeated multiplication of the number 10. In multiplication, \(10^2\) translates to multiplying by '10' twice, resulting in '100'.

• 10¹ equals 10
• 10² equals 100
• 10³ equals 1000

Understanding this concept makes it easier to predict how the decimal will shift when multiplying by powers of 10. As you increase the exponent, you're essentially moving in a larger jump on the place value chart. For every increase in the exponent, an additional zero is added to the result. This intuitive understanding is what allows us to manipulate numbers with ease.

Decimal Point Movement

In multiplication by powers of 10, such as 100, the decimal point movement is a critical concept. Multiplying by \(10\) moves the decimal point one place to the right, hence multiplying by \(100\) shifts it two places to the right. This is because each multiplication by \(10\) shifts the digits toward larger place values.
For instance, beginning with 30.728, each shift changes the decimal through the following steps:

• Multiply by 10: 307.28
• Multiply by another 10: 3072.8

The decimal point jumps past digits sequentially, first '3' to '30', then '30' to '307'. As simple as it sounds, this movement defines the magnitude increase in the number. Children often find it fascinating how the very same digits now mean thousands more in value just by shifting the decimal point.

Place Value Understanding

The concept of place value is foundational in math and helps to make sense of multiplying by powers of 10. In essence, place value refers to the value of each digit in a number based on its position. By understanding place value, one can easily make sense of how numbers change when multiplied.
Consider the number 30.728:

• The digit '3' is in the tens place, giving it a value of 30.
• The digit '7' is in the tenths place, signifying seven-tenths or 0.7.
• During multiplication by 100, a place shift occurs, so '3' becomes 3000, since it's now in the thousands place.

Here's the trick: as digits move to the left across the place values, their worth amplifies tenfold each time. When you multiply by powers of 10, like '100', it reinforces recognition of number growth by leveraging place value.