Convert `(21344)_(5)` to base 10.

Answer

To convert the number \( (21344)_5 \) to base 10, we will use the place value method. Each digit in the base 5 number is multiplied by \( 5 \) raised to the power of its position, starting from 0 on the right.

### Step-by-Step Solution:

1. **Identify the digits and their positions**:
The number \( (21344)_5 \) consists of the digits \( 2, 1, 3, 4, 4 \) from left to right. The positions (from right to left) are 0, 1, 2, 3, and 4.

2. **Write the expression for conversion**:
The base 10 equivalent can be calculated using the formula:
\[ (21344)_5 = 2 \times 5^4 + 1 \times 5^3 + 3 \times 5^2 + 4 \times 5^1 + 4 \times 5^0 \]

3. **Calculate each term**:
- \( 2 \times 5^4 = 2 \times 625 = 1250 \)
- \( 1 \times 5^3 = 1 \times 125 = 125 \)
- \( 3 \times 5^2 = 3 \times 25 = 75 \)
- \( 4 \times 5^1 = 4 \times 5 = 20 \)
- \( 4 \times 5^0 = 4 \times 1 = 4 \)

4. **Sum all the terms**:
Now, we add all the calculated values together:
\[ 1250 + 125 + 75 + 20 + 4 = 1474 \]

5. **Final Result**:
Therefore, the base 10 equivalent of \( (21344)_5 \) is \( 1474 \).