Problem 35 - Algebra and Trigonometry

Question

Perform the indicated operations and write the result in standard form. $$ (-3-\sqrt{-7})^{2} $$

Step-by-Step Solution

Answer

The result of the operation \((-3-\sqrt{-7})^{2}\) in standard form is \(2 + 6i\sqrt{7}\)

Step 1: Rewrite the square root

Rewrite \(\sqrt{-7}\) as \(i\sqrt{7}\), so we get \((-3 - i\sqrt{7})^2\)

Step 2: Apply the binomial formula

Apply the binomial formula for \(a = -3\) and \(b = i\sqrt{7}\) to get \((-3)^2 - 2(-3)(i\sqrt{7}) + (i\sqrt{7})^2\) which simplifies to \(9 + 6i\sqrt{7} - 7\)

Step 3: Simplify the final result

Simplify \(9 + 6i\sqrt{7} - 7\) to get \(2 + 6i\sqrt{7}\).

Step 4: Write the result in standard form

The result in standard form for complex numbers is \(2 + 6i\sqrt{7}\).