Problem 34 - Algebra and Trigonometry

Question

Perform the indicated operations and write the result in standard form. $$ (-5-\sqrt{-9})^{2} $$

Step-by-Step Solution

Answer

The result in standard form is: 34 + 30i.

Step 1: Simplify square root of a negative number

The square root of -9 can be written as \( \sqrt{-9} = \sqrt{9} * \sqrt{-1} = 3i \). Here, 'i' is the imaginary unit.

Step 2: Substitute the simplified square root into the initial expression and square the result

Substitute \( \sqrt{-9} \) with 3i in initial expression. Now it becomes, \( (-5-3i)^{2} \). A binomial square can be calculated as \( (a - b)^2 = a^2 - 2ab + b^2 \). So, \( (-5-3i)^{2} = (-5)^2 -2*(-5)*3i + (3i)^2 = 25 + 30i - 9i^2 \)

Step 3: Convert the result into standard form

Remember that \( i^2 = -1 \). This implies the expression can be further simplified as, 25 + 30i - 9*(-1) = 34 + 30i. This is the standard form of a complex number.