Evaluate \( 3241_{\text {five }}-1342_{\text {five }} \)

To evaluate \( 3241_{\text{five}} - 1342_{\text{five}} \), we'll perform the subtraction directly in base five. ### Step-by-Step Subtraction in Base Five: ``` 3 2 4 1₅ - 1 3 4 2₅ ------------ ``` 1. **Units Place (Rightmost digit):** - **1 - 2:** - Since 1 is less than 2, borrow 1 from the next higher place (fives place). - The 4 in the fives place becomes 3, and the units digit becomes \(1 + 5 = 6\). - Now, \(6 - 2 = 4\). 2. **Fives Place:** - **3 - 4:** - Again, 3 is less than 4, so borrow 1 from the next higher place (25s place). - The 2 in the 25s place becomes 1, and the fives digit becomes \(3 + 5 = 8\). - Now, \(8 - 4 = 4\). 3. **25s Place:** - **1 - 3:** - 1 is less than 3, so borrow 1 from the next higher place (125s place). - The 3 in the 125s place becomes 2, and the 25s digit becomes \(1 + 5 = 6\). - Now, \(6 - 3 = 3\). 4. **125s Place:** - **2 - 1 = 1** ### Final Result: Putting it all together, the result of the subtraction is: \[ 1344_{\text{five}} \] ### Verification in Base Ten: To ensure the subtraction is correct, let's convert both numbers to base ten and subtract: 1. **Convert \(3241_{\text{five}}\) to base ten:** \[ 3 \times 5^3 + 2 \times 5^2 + 4 \times 5^1 + 1 \times 5^0 = 375 + 50 + 20 + 1 = 446 \] 2. **Convert \(1342_{\text{five}}\) to base ten:** \[ 1 \times 5^3 + 3 \times 5^2 + 4 \times 5^1 + 2 \times 5^0 = 125 + 75 + 20 + 2 = 222 \] 3. **Subtract in base ten:** \[ 446 - 222 = 224 \] 4. **Convert \(1344_{\text{five}}\) back to base ten to verify:** \[ 1 \times 5^3 + 3 \times 5^2 + 4 \times 5^1 + 4 \times 5^0 = 125 + 75 + 20 + 4 = 224 \] The verification confirms that the subtraction is correct. **Final Answer:** \[ 3241_{\text{five}} - 1342_{\text{five}} = 1344_{\text{five}} \]