- Evaluate each component of the expression: \(70 + 3 + 0.7 + 0.02 + 0.004\).
- Combine the whole number parts to get \(73\).
- Sum the decimal parts to get \(0.724\).
- Add the results together to reach the final decimal value of \(\boxed{73.724}\).
Explanation
- Analyze the decimal expansion expression
The given expression is a sum of values based on their place value in the decimal system. We have the expression:
\[ (7 \times 10) + (3 \times 1) + \left(7 \times \frac{1}{10}\right) + \left(2 \times \frac{1}{100}\right) + \left(4 \times \frac{1}{1,000}\right) \]
Each term corresponds to a specific place value in a decimal number. The terms represent 7 in the tens place, 3 in the ones place, 7 in the tenths place, 2 in the hundredths place, and 4 in the thousandths place.
- Calculate the value of each term
Now, we evaluate each term individually to convert the expression into a standard decimal form:
- The tens place: \(7 \times 10 = 70\)
- The ones place: \(3 \times 1 = 3\)
- The tenths place: \(7 \times \frac{1}{10} = 0.7\)
- The hundredths place: \(2 \times \frac{1}{100} = 0.02\)
- The thousandths place: \(4 \times \frac{1}{1,000} = 0.004\)
- Sum the values to find the final number
Finally, we add all the evaluated terms together:
\[ 70 + 3 + 0.7 + 0.02 + 0.004 \]
First, sum the whole numbers:
\[ 70 + 3 = 73 \]
Next, sum the decimal parts:
\[ 0.7 + 0.02 + 0.004 = 0.724 \]
Combining these results gives us:
\[ 73 + 0.724 = 73.724 \]
Thus, the base-ten number is \(73.724\).
Examples
Understanding decimal expansion is essential in everyday life, particularly when dealing with currency or precise measurements. For instance, if you are reading a price on a grocery receipt that breaks down items by unit costs, or calculating a physical dimension in meters where each decimal place represents a smaller fraction of the unit (like decimeters, centimeters, and millimeters), you are applying the same principle of summing place values to determine the total value. Being comfortable with these conversions ensures you can accurately interpret quantitative data in any professional or personal setting.