19. Expand $$ (8-\sqrt{3})^{2} $$ giving your answer in form $$ a+b \sqrt{3} $$

Question

UpStudy AI · Accepted Answer

1. Write the expression to be expanded:
   
   $$
   (8 - \sqrt{3})^2
   $$

2. Use the formula for the square of a binomial:
   
   $$
   (a - b)^2 = a^2 - 2ab + b^2
   $$
   
   Here, $$a = 8$$ and $$b = \sqrt{3}$$.

3. Substitute the values into the formula:
   
   $$
   (8 - \sqrt{3})^2 = 8^2 - 2 \cdot 8 \cdot \sqrt{3} + (\sqrt{3})^2
   $$

4. Compute each term:
   
   $$
   8^2 = 64
   $$
   
   $$
   2 \cdot 8 \cdot \sqrt{3} = 16\sqrt{3}
   $$
   
   $$
   (\sqrt{3})^2 = 3
   $$

5. Combine the results:
   
   $$
   64 - 16\sqrt{3} + 3 = 67 - 16\sqrt{3}
   $$

6. Therefore, the expanded form is:
   
   $$
   67 - 16\sqrt{3}
   $$
$$ 67 - 16\sqrt{3} $$

19. Expand \( (8-\sqrt{3})^{2} \) giving your answer in form \( a+b \sqrt{3} \)