Perform the indicated operation. Express your answer in standard form.
${(3+8i)}^2$

• A. $-55+48i$
• B. $9+64i$
• C. $24i$
• D. $9+24i$

Answer: (A)

Verified step by step guidance

1. Recognize that the problem asks you to square the complex number \(\left(3 + 8i\right)^2\). This means you need to multiply \(\left(3 + 8i\right)\) by itself: \(\left(3 + 8i\right) \times \left(3 + 8i\right)\).
2. Use the distributive property (FOIL method) to expand the product: multiply the first terms, outer terms, inner terms, and last terms. Write this as: \$3 \times 3 + 3 \times 8i + 8i \times 3 + 8i \times 8i$.
3. Calculate each multiplication separately: \$3 \times 3 = 9\(, \)3 \times 8i = 24i\(, \)8i \times 3 = 24i\(, and \)8i \times 8i = 64i^2$.
4. Recall that \(i^2 = -1\), so replace \$64i^2\( with \)64 \times (-1) = -64\(. Now combine the real parts and the imaginary parts: real parts are \)9\( and \)-64\(, imaginary parts are \)24i\( and \)24i$.
5. Add the real parts: \$9 - 64\(, and add the imaginary parts: \)24i + 24i\(. Write the final expression in standard form \)a + bi\(, where \)a\( is the real part and \)b\( is the coefficient of \)i$.