Write $1323_4$ in expanded form.

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01-11-2026
Mathematics

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Write $1323_4$ in expanded form.

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-01T10:49:24-05:00

Identify the base $b=4$ and the positional value of each digit $d_i \cdot 4^i$.
Align digits with their respective powers: $1$ at $4^3$, $3$ at $4^2$, $2$ at $4^1$, and $3$ at $4^0$.
Construct the expanded form by summing these products: $1 \cdot 4^3 + 3 \cdot 4^2 + 2 \cdot 4^1 + 3 \cdot 4^0$.
Verify the total as $123$ in base 10; the final expanded form is $\boxed{1 \cdot 4^3 + 3 \cdot 4^2 + 2 \cdot 4^1 + 3 \cdot 4^0}$.

Explanation

Understanding Positional Notation and Base 4

The number $1323_4$ is written in base 4. In any positional numeral system with base $b$, a number represented by the sequence of digits $d_n d_{n-1} \dots d_1 d_0$ has the value $d_n \cdot b^n + d_{n-1} \cdot b^{n-1} + \dots + d_1 \cdot b^1 + d_0 \cdot b^0$. For the number $1323_4$, the base is $b = 4$, and the digits are $d_3=1, d_2=3, d_1=2,$ and $d_0=3$.

Assigning Powers of 4 to Each Digit

To write the number in expanded form, we determine the power of 4 associated with each position, starting from the right (the units place, which corresponds to index 0):

The rightmost digit $3$ is in the $4^0$ position.
The digit $2$ is in the $4^1$ position.
The digit $3$ is in the $4^2$ position.
The leftmost digit $1$ is in the $4^3$ position.

Constructing the Expanded Form

Now, we multiply each digit by its corresponding power of 4 and represent the value as a sum:
$1323_4 = (1 \times 4^3) + (3 \times 4^2) + (2 \times 4^1) + (3 \times 4^0)$
This expression is the expanded form of the number $1323_4$.

Final Expression and Verification

To verify the value in base 10, calculate each component:

$1 \times 4^3 = 1 \times 64 = 64$
$3 \times 4^2 = 3 \times 16 = 48$
$2 \times 4^1 = 2 \times 4 = 8$
$3 \times 4^0 = 3 \times 1 = 3$

Summing these gives $64 + 48 + 8 + 3 = 123$. The expanded form is $\boxed{1 \cdot 4^3 + 3 \cdot 4^2 + 2 \cdot 4^1 + 3 \cdot 4^0}$.

Examples

Expanded form is essential in computer science when working with different number systems, such as binary (base 2) or hexadecimal (base 16). For example, if a computer stores a color code as a base-16 number, understanding the expanded form allows you to convert that color code into a standard decimal (base 10) value that identifies the specific intensity of red, green, and blue components. By mastering the expanded form, you gain a foundational skill for understanding how data is structured and processed at the machine level.