Problem 73 - Prealgebra

Question

Divide the decimal by the given power of 10 . \(\frac{341.16}{10^{3}}\)

Step-by-Step Solution

Answer

0.34116

Step 1: Identify the Power of 10

The problem asks us to divide 341.16 by a power of 10, specifically by \(10^3\). This means we are dividing by 1000 since \(10^3 = 1000\).

Step 2: Shift the Decimal Point

To divide 341.16 by 1000, move the decimal point three places to the left. This is because 1000 (or \(10^3\)) has three zeros.

Step 3: Division

Moving the decimal point three places to the left from 341.16 (where it is between 1 and 6) gives us 0.34116. Therefore, \(\frac{341.16}{1000} = 0.34116\).

Key Concepts

Powers of Ten

Understanding powers of ten is essential for tackling problems involving the division of decimal numbers by powers like 10, 100, 1000, etc. A power of ten is simply ten raised to an exponent, represented as \(10^n\), where \(n\) is a whole number. The exponent tells us how many times to multiply 10 by itself. For example, \(10^1 = 10\) and \(10^3 = 1000\).
When you divide by a power of ten, such as \(10^3\), you are using the value of 1000 because \(10^3 = 1000\). This division effectively helps simplify decimal division to a matter of moving decimal points.

• Multiplying by \(10^n\) shifts the decimal point to the right \(n\) places.
• Dividing by \(10^n\) shifts the decimal point to the left \(n\) places.

This understanding allows arithmetic manipulations without long division, making it easier for quick calculations using just your understanding of powers of ten.

Decimal Point

The position of the decimal point is crucial in decimal arithmetic. It signifies the boundary between whole numbers and fractions.
When dividing a decimal by a power of ten, understanding how and where to shift the decimal point becomes key to solving the problem.• Shifting the decimal point to the right increases the value of the number. This is the same as multiplying by powers of ten.
• When you move the decimal point to the left, you are effectively reducing the number, which is equivalent to division by powers of ten.
Let’s take the example given, \(341.16\). To divide it by \(10^3\), or 1000, all we do is shift the decimal point three places to the left.

• Current decimal point: between 1 and 6 \( (341.16)\).
• New decimal point: before 3, resulting in \(0.34116\).

Thus, the decimal shifts provide a way to transform the number into a quotient with reduced value.

Step-by-Step Solutions

Approaching a problem with a step-by-step solution can simplify complex arithmetic operations involving decimals. Here’s a streamlined way to deal with the division of decimals by powers of ten like the original exercise.
**Step 1: Know Your Power of Ten**
Identify the power of ten you are working with. In the example, it’s \(10^3\). Translates into a real number with zeros, namely 1000 in this case.

**Step 2: Shift the Decimal Point**
Count the zeros in the power of ten, which tells you how many places to move the decimal point. Here, moving the decimal three spaces to the left efficiently divides by 1000.

**Step 3: Verify the Calculation**
After shifting the decimal point for \(341.16\), the result is \(0.34116\). Always check if the decimal was shifted correctly by confirming the quotient makes sense in terms of size and decimal positioning.

• Was the number appropriately reduced in value?
• Is the decimal represented correctly in the results?

Such detailed frameworks help students confidently divide decimals using powers of ten.