Question

lesson activity
adding, subtracting, and multiplying complex numbers
this activity will help you meet these educational goals:
you will practice adding, subtracting, and multiplying complex numbers.
directions
read the instructions for this self - checked activity. type in your response to each question, and check your answers. at the end of the activity, write a brief evaluation of your work.
activity
in this activity, you will add, subtract, and multiply complex numbers to solve more complicated problems.
question 1
enter the correct answer in the box.
write this expression as a complex number in standard form, (a + bi).
((-3sqrt{-81})(-5+sqrt{-9})+9i)

Explanation:

Step1: Simplify square roots of negative numbers

Recall that $\sqrt{-a} = i\sqrt{a}$ for $a>0$. So, $\sqrt{-81} = i\sqrt{81}=9i$ and $\sqrt{-9}=i\sqrt{9} = 3i$. Substitute these into the expression:
$(-3\sqrt{-81})(-5+\sqrt{-9})+9i=(-3\times9i)(-5 + 3i)+9i$

Step2: Simplify the first product

Multiply the coefficients and the complex numbers: $(-27i)(-5 + 3i)+9i$. Using the distributive property $a(b + c)=ab+ac$, we get $(-27i)\times(-5)+(-27i)\times(3i)+9i$.
Calculate each term: $135i-81i^{2}+9i$. Recall that $i^{2}=-1$, so $-81i^{2}=-81\times(-1) = 81$.

Step3: Combine like terms

Combine the $i$ terms and the constant term: $81+(135i + 9i)=81 + 144i$.

Answer:

$81 + 144i$