7. $(-3 + 2i)(4 + 8i)$
8. $(-3 - 5i)^2$
9. $(2 - i)(2 + i)$
10. $(4 + \sqrt{-48}) - (8 - \sqrt{-24})$
11. $5(9 + \sqrt{-20}) + (15 - \sqrt{-45})$
12. $3i(10 - \sqrt{-24}) - i(4 + \sqrt{-294})$

Question

Accepted Answer

### Question 7: $(-3 + 2i)(4 + 8i)$
# Explanation:
## Step 1: Apply the distributive property (FOIL method)
We multiply each term in the first complex number by each term in the second complex number:
$$
\begin{align*}
(-3)(4) + (-3)(8i) + (2i)(4) + (2i)(8i) &= -12 - 24i + 8i + 16i^2
\end{align*}
$$
## Step 2: Simplify using $i^2 = -1$
Recall that $i^2 = -1$, so we substitute that in and combine like terms:
$$
\begin{align*}
-12 - 24i + 8i + 16(-1) &= -12 - 24i + 8i - 16\
&= (-12 - 16) + (-24i + 8i)\
&= -28 - 16i
\end{align*}
$$
# Answer:
$-28 - 16i$

### Question 8: $(-3 - 5i)^2$
# Explanation:
## Step 1: Expand the square using the formula $(a - b)^2 = a^2 - 2ab + b^2$ (here $a = -3$, $b = 5i$)
$$
\begin{align*}
(-3)^2 - 2(-3)(5i) + (5i)^2 &= 9 + 30i + 25i^2
\end{align*}
$$
## Step 2: Simplify using $i^2 = -1$
Substitute $i^2 = -1$ and simplify:
$$
\begin{align*}
9 + 30i + 25(-1) &= 9 + 30i - 25\
&= (9 - 25) + 30i\
&= -16 + 30i
\end{align*}
$$
# Answer:
$-16 + 30i$

### Question 9: $(2 - i)(2 + i)$
# Explanation:
## Step 1: Recognize the difference of squares formula $(a - b)(a + b) = a^2 - b^2$ (here $a = 2$, $b = i$)
$$
\begin{align*}
2^2 - i^2
\end{align*}
$$
## Step 2: Simplify using $i^2 = -1$
Substitute $i^2 = -1$:
$$
\begin{align*}
4 - (-1) &= 4 + 1\
&= 5
\end{align*}
$$
# Answer:
$5$

### Question 10: $(4 + \sqrt{-48}) - (8 - \sqrt{-24})$
# Explanation:
## Step 1: Simplify the square roots of negative numbers
Recall that $\sqrt{-a} = i\sqrt{a}$ for $a > 0$:
$$
\begin{align*}
\sqrt{-48} &= i\sqrt{48} = i\sqrt{16 \times 3} = 4i\sqrt{3}\
\sqrt{-24} &= i\sqrt{24} = i\sqrt{4 \times 6} = 2i\sqrt{6}
\end{align*}
$$
## Step 2: Substitute back into the expression and simplify
$$
\begin{align*}
(4 + 4i\sqrt{3}) - (8 - 2i\sqrt{6}) &= 4 + 4i\sqrt{3} - 8 + 2i\sqrt{6}\
&= (4 - 8) + 4i\sqrt{3} + 2i\sqrt{6}\
&= -4 + 4i\sqrt{3} + 2i\sqrt{6}
\end{align*}
$$
# Answer:
$-4 + 4i\sqrt{3} + 2i\sqrt{6}$

### Question 11: $5(9 + \sqrt{-20}) + (15 - \sqrt{-45})$
# Explanation:
## Step 1: Simplify the square roots of negative numbers
Recall that $\sqrt{-a} = i\sqrt{a}$ for $a > 0$:
$$
\begin{align*}
\sqrt{-20} &= i\sqrt{20} = i\sqrt{4 \times 5} = 2i\sqrt{5}\
\sqrt{-45} &= i\sqrt{45} = i\sqrt{9 \times 5} = 3i\sqrt{5}
\end{align*}
$$
## Step 2: Distribute and simplify
$$
\begin{align*}
5(9) + 5(2i\sqrt{5}) + 15 - 3i\sqrt{5} &= 45 + 10i\sqrt{5} + 15 - 3i\sqrt{5}\
&= (45 + 15) + (10i\sqrt{5} - 3i\sqrt{5})\
&= 60 + 7i\sqrt{5}
\end{align*}
$$
# Answer:
$60 + 7i\sqrt{5}$

### Question 12: $3i(10 - \sqrt{-24}) - i(4 + \sqrt{-294})$
# Explanation:
## Step 1: Simplify the square roots of negative numbers
Recall that $\sqrt{-a} = i\sqrt{a}$ for $a > 0$:
$$
\begin{align*}
\sqrt{-24} &= i\sqrt{24} = i\sqrt{4 \times 6} = 2i\sqrt{6}\
\sqrt{-294} &= i\sqrt{294} = i\sqrt{49 \times 6} = 7i\sqrt{6}
\end{align*}
$$
## Step 2: Distribute and simplify
$$
\begin{align*}
3i(10) - 3i(2i\sqrt{6}) - i(4) - i(7i\sqrt{6}) &= 30i - 6i^2\sqrt{6} - 4i - 7i^2\sqrt{6}
\end{align*}
$$
## Step 3: Simplify using $i^2 = -1$
Substitute $i^2 = -1$ and combine like terms:
$$
\begin{align*}
30i - 6(-1)\sqrt{6} - 4i - 7(-1)\sqrt{6} &= 30i + 6\sqrt{6} - 4i + 7\sqrt{6}\
&= (30i - 4i) + (6\sqrt{6} + 7\sqrt{6})\
&= 26i + 13\sqrt{6}
\end{align*}
$$
# Answer:
$13\sqrt{6} + 26i$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Question 7: $(-3 + 2i)(4 + 8i)$

Step 1: Apply the distributive property (FOIL method)

Step 2: Simplify using $i^2 = -1$

Step 1: Expand the square using the formula $(a - b)^2 = a^2 - 2ab + b^2$ (here $a = -3$, $b = 5i$)

Step 2: Simplify using $i^2 = -1$

Step 1: Recognize the difference of squares formula $(a - b)(a + b) = a^2 - b^2$ (here $a = 2$, $b = i$)

Step 2: Simplify using $i^2 = -1$

Answer:

Question 8: $(-3 - 5i)^2$