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 without simplifying the final numeric values: First, \((4\sqrt{3})^2\) means square both 4 and \(\sqrt{3}\), so it becomes \(4^2 \times (\sqrt{3})^2\).
4. Next, calculate the middle term \(-2 \times (4\sqrt{3}) \times 5\) by multiplying the constants and keeping the square root as is.
5. Finally, calculate the last term \$5^2$. Then combine all three terms together to write the expanded expression before simplifying.