Verified Answer :

  • Identify the place value for each term in the given expressions, where each power of ten \(10^n\) corresponds to a specific digit position.
  • Fill in any missing powers of ten with the digit 0.
  • Sum the components to arrive at the standard base-ten numeral.
  • The results are \(8840\) and \(5000042\).
  • Final Answer: \(\boxed{8840}, \boxed{5000042}\)

Explanation

  1. Analysis of Base-Ten Notation

In base-ten notation, any number can be expressed as a sum of powers of ten multiplied by a digit (0-9). The position of the digit determines its place value. For example, a number can be represented as \(d_n \cdot 10^n + d_{n-1} \cdot 10^{n-1} + \dots + d_1 \cdot 10^1 + d_0 \cdot 10^0\). If a power of ten is missing from the expression, it means the corresponding digit in that place value is 0. We will evaluate each expression by calculating these products and summing them to find the standard numeral.

  1. Converting Expression a

For expression (a), we have \(8 \cdot 10^3 + 8 \cdot 10^2 + 4 \cdot 10\). We identify the place values involved:

  • The term \(8 \cdot 10^3\) represents 8 in the thousands place.
  • The term \(8 \cdot 10^2\) represents 8 in the hundreds place.
  • The term \(4 \cdot 10\) (or \(4 \cdot 10^1\)) represents 4 in the tens place.
  • There is no term for \(10^0\), so the ones place is 0.

Summing these, we get \(8000 + 800 + 40 + 0 = 8840\).

  1. Converting Expression b

For expression (b), we have \(5 \cdot 10^6 + 4 \cdot 10 + 2\). We identify the place values involved:

  • The term \(5 \cdot 10^6\) represents 5 in the millions place.
  • Powers of ten between \(10^5\) and \(10^2\) are missing, so these places are filled with 0.
  • The term \(4 \cdot 10\) represents 4 in the tens place.
  • The term \(2\) (or \(2 \cdot 10^0\)) represents 2 in the ones place.

Placing these into the standard format, we have 5 in the millions place, 0 in the hundred-thousands, ten-thousands, thousands, and hundreds places, 4 in the tens place, and 2 in the ones place. This gives us 5,000,042.

  1. Conclusion

Combining the results, the base-ten numerals for the given expressions are:

A\(8 \cdot 10^3 + 8 \cdot 10^2 + 4 \cdot 10 = \boxed{8840}\)
B\(5 \cdot 10^6 + 4 \cdot 10 + 2 = \boxed{5000042}\)

Examples

In everyday life, we use base-ten numerals to represent quantities such as currency, measurements, or population counts. For instance, if you are counting money and have five \$1,000 bills, eight \$100 bills, and four \$10 bills, the total amount is \(5 \cdot 1000 + 8 \cdot 100 + 4 \cdot 10 = 5000 + 800 + 40 = \$5840\). Converting expanded expressions into standard numerals helps us easily compare numbers, perform calculations, and understand the magnitude of values in various real-world situations.

Related Questions