Question

1. simplify.

a. $sqrt{-49}=7i$
b. $6sqrt{-12}$
c. $i^{90}$
d. $i^{13}$
e. $(7 - 6i)+(9 + 11i)$
f. $(5 + 2i)-(-13 + 4i)$
g. $(2 - 6i)^{2}$
h. $(sqrt{5}+2i)^{2}$
i. $\frac{6 + 5i}{-2i}$

Explanation:

Step1: Simplify $\sqrt{-12}$

$6\sqrt{-12}=6\sqrt{12}\times\sqrt{-1}=6\times2\sqrt{3}i = 12\sqrt{3}i$

Step2: Use $i^n$ cycle property for $i^{90}$

Since $i^1 = i$, $i^2=- 1$, $i^3=-i$, $i^4 = 1$ and $90\div4 = 22\cdots\cdots2$. So $i^{90}=(i^4)^{22}\times i^2=1^{22}\times(-1)=-1$

Step3: Use $i^n$ cycle property for $i^{13}$

$13\div4 = 3\cdots\cdots1$. So $i^{13}=(i^4)^3\times i^1=1^3\times i = i$

Step4: Combine like - terms for $(7 - 6i)+(9 + 11i)$

$(7+9)+(-6 + 11)i=16 + 5i$

Step5: Distribute and combine like - terms for $(5 + 2i)-(-13 + 4i)$

$5 + 2i+13-4i=(5 + 13)+(2-4)i=18-2i$

Step6: Expand $(2 - 6i)^2$ using $(a - b)^2=a^2-2ab + b^2$

$(2 - 6i)^2=2^2-2\times2\times6i+(6i)^2=4-24i + 36i^2=4-24i-36=-32-24i$

Step7: Expand $(\sqrt{5}+2i)^2$ using $(a + b)^2=a^2+2ab + b^2$

$(\sqrt{5}+2i)^2=(\sqrt{5})^2+2\times\sqrt{5}\times2i+(2i)^2=5 + 4\sqrt{5}i-4=1 + 4\sqrt{5}i$

Step8: Rationalize the denominator for $\frac{6 + 5i}{-2i}$

Multiply numerator and denominator by $i$: $\frac{(6 + 5i)\times i}{-2i\times i}=\frac{6i+5i^2}{-2i^2}=\frac{6i - 5}{2}=-\frac{5}{2}+3i$

Answer:

a. $7i$
b. $12\sqrt{3}i$
c. $-1$
d. $i$
e. $16 + 5i$
f. $18-2i$
g. $-32-24i$
h. $1 + 4\sqrt{5}i$
i. $-\frac{5}{2}+3i$