Answered

Write $4023_{\text {six }}$ in expanded notation and then convert it to base ten.



Write the number $4023_{\text {six }}$ in expanded notation. Choose the correct answer below.

A$4 \cdot 6^3+0 \cdot 6^2+2 \cdot 6^1+3$
B$4 \cdot 6+0 \cdot 6+2 \cdot 6+3 \cdot 6$
C$4 \cdot 6^4+0 \cdot 6^3+2 \cdot 6^2+3 \cdot 6^1$
D$4 \cdot 6^2+0 \cdot 6^1+2 \cdot 6^0+3$



The value of $4023_{\text {six }}$ in base ten is □ (Simplify your answer.)

Verified Answer :

  • Write the number \(4023_{\text{six}}\) in expanded notation by assigning powers of 6 to each digit position: \(4 \cdot 6^3 + 0 \cdot 6^2 + 2 \cdot 6^1 + 3 \cdot 6^0\).
  • Identify that this matches option A: \(4 \cdot 6^3 + 0 \cdot 6^2 + 2 \cdot 6^1 + 3\).
  • Calculate the decimal equivalent by evaluating each term: \(864 + 0 + 12 + 3 = 879\).
  • The value in base ten is \(\boxed{879}\).

Explanation

  1. Expanded Notation Analysis

To express the number \(4023_{\text{six}}\) in expanded notation, we identify the place value of each digit relative to the base, which is \(6\). A number represented in base \(b\) as \(d_n d_{n-1} \dots d_1 d_0\) corresponds to the value \(\sum_{i=0}^{n} d_i \cdot b^i\). For our number \(4023_{\text{six}}\), we have the following digits and their corresponding positions:

  • Digit \(4\) is in the position \(3\), corresponding to \(6^3\).
  • Digit \(0\) is in the position \(2\), corresponding to \(6^2\).
  • Digit \(2\) is in the position \(1\), corresponding to \(6^1\).
  • Digit \(3\) is in the position \(0\), corresponding to \(6^0\).
  1. Identifying the Expanded Form

Based on the positions identified, we write the expanded notation as follows:
\[4 \cdot 6^3 + 0 \cdot 6^2 + 2 \cdot 6^1 + 3 \cdot 6^0\]
Since \(6^0 = 1\), this expression simplifies to:
\[4 \cdot 6^3 + 0 \cdot 6^2 + 2 \cdot 6^1 + 3\]
This matches option A.

  1. Calculating the Base Ten Value

To convert the number to base ten, we compute the value of each term in the expanded notation:

  • The first term is \(4 \cdot 6^3 = 4 \cdot 216 = 864\).
  • The second term is \(0 \cdot 6^2 = 0 \cdot 36 = 0\).
  • The third term is \(2 \cdot 6^1 = 2 \cdot 6 = 12\).
  • The fourth term is \(3 \cdot 6^0 = 3 \cdot 1 = 3\).

Summing these values gives:
\[864 + 0 + 12 + 3 = 879\]
Thus, the value of \(4023_{\text{six}}\) in base ten is \(879\).

Examples

Understanding base systems is essential for computer science and digital electronics. Computers perform internal operations using binary (base 2), while programmers often use hexadecimal (base 16) or octal (base 8) to represent binary data more compactly. For instance, color codes on websites use hexadecimal notation (like #FFFFFF for white), where each digit represents a specific power of 16. By learning to convert between different bases, you grasp how data is stored and manipulated at the most fundamental level of technology.

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