If 54\(_{ten}\) = X\(_{four}\), find the value of X to 3 decimal point

Answer Details

To find which number in base 4 is equal to 54 in base 10, we need to convert the decimal number 54 into base 4.

Step 1: Understanding Base 4
In base 4, each digit represents a power of 4. For example, a three-digit number \(abc_{four}\) means:

• \(a \times 4^2\)
• \(+ b \times 4^1\)
• \(+ c \times 4^0\)

Step 2: Convert 54 from Decimal to Base 4
We need to express 54 as a sum of powers of 4 with coefficients less than 4.

• \(4^3 = 64\) is too large (since 54 < 64).
• \(4^2 = 16\) is the highest power we can use.

Now, divide 54 by 16: \[ \frac{54}{16} = 3.375 \] So, the coefficient for \(4^2\) is 3.
\(3 \times 16 = 48\).

Subtract this from 54: \[ 54 - 48 = 6 \]

Now, use \(4^1 = 4\): \[ \frac{6}{4} = 1.5 \] So, the coefficient for \(4^1\) is 1.
\(1 \times 4 = 4\).

Subtract this from 6: \[ 6 - 4 = 2 \]

Now, use \(4^0 = 1\): \[ \frac{2}{1} = 2 \] So, the coefficient for \(4^0\) is 2.

Putting it all together, the base 4 representation is: \[ 3 \times 4^2 + 1 \times 4^1 + 2 \times 4^0 = 48 + 4 + 2 = 54 \] So, \(54_{ten} = 312_{four}\).

Why this is correct:
By dividing 54 by decreasing powers of 4 and assigning the appropriate digits, we guarantee that every digit is less than 4, and the sum matches the original decimal value.

The correct answer is the number 312 written in base 4.