Simplify the following.
$(4\sqrt{3}-5{)}^2$

• A. $73-40\sqrt{3}$
• B. $-9-20\sqrt{3}$
• C. $41-20\sqrt{3}$
• D. $23-40\sqrt{3}$

Answer: (A)

Verified step by step guidance

1. Recognize that the expression is a binomial squared: \((4\sqrt{3} - 5)^2\). Use the formula for squaring a binomial: \((a - b)^2 = a^2 - 2ab + b^2\).
2. Identify \(a = 4\sqrt{3}\) and \(b = 5\). Substitute these into the formula: \((4\sqrt{3})^2 - 2 \times (4\sqrt{3}) \times 5 + 5^2\).
3. Calculate each term separately: First, \((4\sqrt{3})^2 = 4^2 \times (\sqrt{3})^2 = 16 \times 3\).
4. Next, calculate the middle term: \(-2 \times 4\sqrt{3} \times 5 = -40\sqrt{3}\).
5. Finally, calculate the last term: \$5^2 = 25$. Then combine all terms: \(16 \times 3 - 40\sqrt{3} + 25\).