Find the unit digit in `(6 ^(36))^(84).`

Options

A. 6
B. 4
C. 7
D. 1

Answer

To find the unit digit of \((6^{36})^{84}\), we can follow these steps:

### Step 1: Simplify the Expression
The expression can be simplified using the property of exponents:
\[ (6^{36})^{84} = 6^{36 \times 84} \]

### Step 2: Calculate the Exponent
Now, calculate the exponent:
\[ 36 \times 84 = 3024 \]
So, we need to find the unit digit of \(6^{3024}\).

### Step 3: Identify the Pattern of Unit Digits
Next, we observe the unit digits of powers of 6:
- \(6^1 = 6\) (unit digit is 6)
- \(6^2 = 36\) (unit digit is 6)
- \(6^3 = 216\) (unit digit is 6)
- \(6^4 = 1296\) (unit digit is 6)

From this, we can see that the unit digit of \(6^n\) for any positive integer \(n\) is always 6.

### Step 4: Conclusion
Thus, the unit digit of \(6^{3024}\) is 6.

Therefore, the unit digit in \((6^{36})^{84}\) is **6**.