To simplify the multiplication of \(213_4\) and \(23_4\), we will first convert both numbers from base 4 to base 10, perform the multiplication in base 10, and then convert the result back to base 4. ### Step 1: Convert \(213_4\) to Base 10 The number \(213_4\) can be converted to base 10 using the formula: \[ n = d_k \cdot b^k + d_{k-1} \cdot b^{k-1} + \ldots + d_1 \cdot b^1 + d_0 \cdot b^0 \] where \(d\) are the digits of the number, \(b\) is the base (which is 4 in this case), and \(k\) is the position of the digit from the right (starting from 0). For \(213_4\): - The digit \(2\) is in the \(4^2\) place. - The digit \(1\) is in the \(4^1\) place. - The digit \(3\) is in the \(4^0\) place. Calculating each term: \[ 2 \cdot 4^2 = 2 \cdot 16 = 32 \] \[ 1 \cdot 4^1 = 1 \cdot 4 = 4 \] \[ 3 \cdot 4^0 = 3 \cdot 1 = 3 \] Now, add these values together: \[ 32 + 4 + 3 = 39 \] So, \(213_4 = 39_{10}\). ### Step 2: Convert \(23_4\) to Base 10 Now, we convert \(23_4\) to base 10 using the same method: For \(23_4\): - The digit \(2\) is in the \(4^1\) place. - The digit \(3\) is in the \(4^0\) place. Calculating each term: \[ 2 \cdot 4^1 = 2 \cdot 4 = 8 \] \[ 3 \cdot 4^0 = 3 \cdot 1 = 3 \] Now, add these values together: \[ 8 + 3 = 11 \] So, \(23_4 = 11_{10}\). ### Step 3: Multiply the Base 10 Values Now we multiply the two base 10 values we obtained: \[ 39 \times 11 = 429 \] ### Step 4: Convert \(429_{10}\) Back to Base 4 To convert \(429_{10}\) back to base 4, we repeatedly divide by 4 and keep track of the remainders: 1. \(429 \div 4 = 107\) remainder \(1\) 2. \(107 \div 4 = 26\) remainder \(3\) 3. \(26 \div 4 = 6\) remainder \(2\) 4. \(6 \div 4 = 1\) remainder \(2\) 5. \(1 \div 4 = 0\) remainder \(1\) Now, we read the remainders from bottom to top to get the base 4 representation: Thus, \(429_{10} = 12321_4\). ### Step 5: Check the Options Now, let's compare our result \(12321_4\) with the provided options: A. \(103114\) B. \(103214\) C. \(122314\) D. \(132114\) None of the options match \(12321_4\). It seems there was a misunderstanding in the options provided. ### Conclusion The correct answer for the multiplication of \(213_4\) and \(23_4\) is \(12321_4\), which is not listed among the options. ### Revision Summary - Convert numbers from base 4 to base 10 using positional values. - Multiply the base 10 results. - Convert the product back to base 4 using repeated division and remainders. - Always verify your final answer against the provided options. If you encounter discrepancies, double-check your calculations and ensure the options are correct.