Verified Answer :

  • Evaluate individual powers of 10: \(1 \cdot 10^6 = 1,000,000\), \(5 \cdot 10^4 = 50,000\), and \(4 \cdot 10^1 = 40\).
  • Align the resulting values by their place value to ensure proper summation.
  • Combine these values into a single decimal numeral: \(1,000,000 + 50,000 + 40 = 1,050,040\).
  • The final base-ten numeral is \(\boxed{1050040}\).

Explanation

  1. Analyzing the Expression

To convert the expression \(1 \cdot 10^6 + 5 \cdot 10^4 + 4 \cdot 10^1\) into a standard base-ten numeral, we first need to evaluate each of the powers of ten. The given expression is composed of three separate terms representing different place values in the decimal system. We analyze each term individually to determine its numerical contribution to the final sum.

  1. Evaluating the Powers of Ten

We calculate the value of each term:

  • The first term is \(1 \cdot 10^6\), which is \(1 \cdot 1,000,000 = 1,000,000\).
  • The second term is \(5 \cdot 10^4\), which is \(5 \cdot 10,000 = 50,000\).
  • The third term is \(4 \cdot 10^1\), which is \(4 \cdot 10 = 40\).
  1. Summing the Terms and Finalizing the Numeral

Now, we add these results together:
\[1,000,000 + 50,000 + 40\]
To ensure the correct placement of each digit in the base-ten system, we align the place values:

  • The millions place is \(1\).
  • The hundred-thousands place is \(0\).
  • The ten-thousands place is \(5\).
  • The thousands place is \(0\).
  • The hundreds place is \(0\).
  • The tens place is \(4\).
  • The ones place is \(0\).

Combining these digits, we arrive at the standard base-ten numeral: \(1,050,040\).

Examples

Scientific notation is widely used in science and engineering to represent very large or very small numbers compactly. For example, the distance from the Earth to the Sun is approximately \(1.496 \times 10^8\) kilometers, and the mass of the Earth is about \(5.97 \times 10^{24}\) kilograms. Converting these into standard numerals helps in practical applications, like calculating total travel distances, fuel requirements for space exploration, or data storage capacities in computing, where precise base-ten representation is often necessary for standard record-keeping and processing.

Related Questions