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 · 2026-11-01T10:50:28-05:00

Identify the base $b=4$ and the digits of the number $1323_4$ as $1, 3, 2, 3$.
Assign each digit its correct place value power of 4, starting from the rightmost digit at $4^0$.
Multiply each digit by its corresponding power and sum them to obtain the expanded form.
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

A number in base $b$ is written as a sequence of digits $d_n d_{n-1} \dots d_1 d_0$. Its value is represented by the sum of its digits multiplied by the corresponding power of the base, expressed as:
\[\sum_{i=0}^{n} d_i \cdot b^i = d_n \cdot b^n + d_{n-1} \cdot b^{n-1} + \dots + d_1 \cdot b^1 + d_0 \cdot b^0\]
In the number $1323_4$, the base $b$ is $4$. The digits provided are $d_3=1$, $d_2=3$, $d_1=2$, and $d_0=3$, reading from left to right.

Assigning Place Values

To write the number in expanded form, we assign each digit to its corresponding power of the base $4$, starting from the rightmost digit which corresponds to $4^0$:

The rightmost digit is $3$, which occupies the position of $4^0$.
The next digit to the left is $2$, which occupies the position of $4^1$.
The next digit to the left is $3$, which occupies the position of $4^2$.
The leftmost digit is $1$, which occupies the position of $4^3$.

Formulating the Expanded Expression

By multiplying each digit by its respective power of the base and summing them together, we get the expanded form:
\[1 \cdot 4^3 + 3 \cdot 4^2 + 2 \cdot 4^1 + 3 \cdot 4^0\]
This expression clearly shows how the value of each digit is determined by its position in the base-4 system.

Final Answer

The expanded form of the number $1323_4$ is given by the expression:
\[\boxed{1 \cdot 4^3 + 3 \cdot 4^2 + 2 \cdot 4^1 + 3 \cdot 4^0}\]

Examples

Understanding expanded form is crucial in computer science and digital systems where data is often represented in binary (base 2), octal (base 8), or hexadecimal (base 16). For example, a color code in hexadecimal, like $\#FFFFFF$, is interpreted by breaking down the digits into powers of 16 to determine the intensity of red, green, and blue. Just as we expanded $1323_4$ into powers of 4, programmers expand hexadecimal codes into powers of 16 to translate human-readable codes into machine-readable instructions.