Find out the unit digit in `(12^(55))/(3^(11))+(8^(48))/(16^(18))`

Options

A. 2
B. 4
C. 0
D. 6

Answer

To find the unit digit of the expression \((12^{55})/(3^{11}) + (8^{48})/(16^{18})\), we can simplify and analyze each term step by step.

### Step 1: Simplify the first term \((12^{55})/(3^{11})\)

We can rewrite \(12\) as \(3 \times 4\):
\[ 12^{55} = (3 \times 4)^{55} = 3^{55} \times 4^{55} \]
Thus, the first term becomes:
\[ \frac{12^{55}}{3^{11}} = \frac{3^{55} \times 4^{55}}{3^{11}} = 3^{55 - 11} \times 4^{55} = 3^{44} \times 4^{55} \]

### Step 2: Simplify the second term \((8^{48})/(16^{18})\)

We can rewrite \(8\) and \(16\) as powers of \(2\):
\[ 8^{48} = (2^3)^{48} = 2^{144} \]
\[ 16^{18} = (2^4)^{18} = 2^{72} \]
Thus, the second term becomes:
\[ \frac{8^{48}}{16^{18}} = \frac{2^{144}}{2^{72}} = 2^{144 - 72} = 2^{72} \]

### Step 3: Combine the simplified terms

Now we have:
\[ 3^{44} \times 4^{55} + 2^{72} \]

### Step 4: Find the unit digit of each term

1. **Finding the unit digit of \(3^{44}\)**:
The unit digits of powers of \(3\) cycle every \(4\):
- \(3^1 \equiv 3\)
- \(3^2 \equiv 9\)
- \(3^3 \equiv 7\)
- \(3^4 \equiv 1\)

Since \(44 \mod 4 = 0\), the unit digit of \(3^{44}\) is \(1\).

2. **Finding the unit digit of \(4^{55}\)**:
The unit digits of powers of \(4\) cycle every \(2\):
- \(4^1 \equiv 4\)
- \(4^2 \equiv 6\)

Since \(55 \mod 2 = 1\), the unit digit of \(4^{55}\) is \(4\).

3. **Finding the unit digit of \(2^{72}\)**:
The unit digits of powers of \(2\) cycle every \(4\):
- \(2^1 \equiv 2\)
- \(2^2 \equiv 4\)
- \(2^3 \equiv 8\)
- \(2^4 \equiv 6\)

Since \(72 \mod 4 = 0\), the unit digit of \(2^{72}\) is \(6\).

### Step 5: Combine the unit digits

Now we can calculate the unit digit of the entire expression:
- The unit digit of \(3^{44} \times 4^{55}\) is \(1 \times 4 = 4\).
- The unit digit of \(2^{72}\) is \(6\).

Finally, we add these unit digits:
\[ 4 + 6 = 10 \]
The unit digit of \(10\) is \(0\).

### Final Answer
The unit digit of the expression \((12^{55})/(3^{11}) + (8^{48})/(16^{18})\) is **0**.