Problem 67 - Prealgebra

Question

Divide the decimal by the given power of 10 . \(\frac{563.94}{10^{3}}\)

Step-by-Step Solution

Answer

The result is 0.56394.

Step 1: Identify the Power of 10

The problem asks us to divide the decimal 563.94 by the power of 10, which is \(10^3\). This means we need to move the decimal point 3 places to the left because \(10^3 = 1000\).

Step 2: Shift the Decimal Point

Starting from the decimal point in 563.94, move it three places to the left. This converts the original number into a smaller number.

Step 3: Placeholding Zeroes

After shifting the decimal, if there are any missing places before the number, you must add zeros to maintain numerical places. In this case, moving the decimal three places left turns 563.94 into 0.56394.

Step 4: Express the Result

The final result of dividing 563.94 by \(10^3\) is 0.56394. This number is 1/1000 of the original number.

Key Concepts

Powers of Ten

Understanding powers of ten is crucial when it comes to dividing decimals. A power of ten is simply a number that is 10 raised to an exponent. For example, in the expression \(10^3\), the exponent is 3. This means you multiply 10 by itself three times, resulting in 1000. Powers of ten help us transform numbers in a predictable way by changing the position of the decimal point.

When you divide a number by a power of ten, you move the decimal point to the left as many places as the value of the exponent. This is essentially making the number smaller by factors of ten. It’s like cutting a pizza into smaller slices to share it more broadly!

• \(10^1\) indicates moving the decimal one place.
• \(10^2\) results in moving it two places.
• \(10^3\) signifies moving it three places, and so on.

By shifting the decimal, the number decreases in size, consistent with dividing by larger and larger powers of ten.

Decimal Point

A decimal point is a dot that separates the whole number part from the fractional part in a decimal number. Understanding its placement and shifting is a big part of working with decimals, especially when dividing by powers of ten.

When you divide by 10, you shift the decimal point one place to the left. Let’s see how it works with our example number, 563.94. By dividing by \(10^3\), which equals 1000, you're required to shift the decimal three spaces to the left. This adjusts 563.94 to become 0.56394, essentially spreading the value across a larger scale - from hundreds down to thousandths.

• The number to the left of the decimal is the whole number part.
• Numbers to the right represent the fraction of the whole.

Understanding this movement is vital for accurately scaling numbers and performing calculations involving decimals.

Place Value

Place value helps identify the value of each digit in a number based on its position. In a decimal number, this concept is vital as it relates to both whole numbers and fractional parts.

When you move the decimal point in a number, you change its place value. In 563.94:

• "5" is in the hundreds place.
• "6" is in the tens place.
• "3" is in the ones place.
• "9" is in the tenths place.
• "4" is in the hundredths place.

After shifting the decimal point three places to the left, the place values change significantly:

• Now "5" is in the tenths place.
• "6" is in the hundredths place.
• "3" is in the thousandths place.

Place value makes clear how shifting the decimal point alters the value of the entire number, reinforcing the effect of dividing by powers of ten in terms of expanding or contracting numerical size. Understanding these place transitions ensures accuracy in your calculations.