Question

rationalizing imaginary denominators
simplify.

1. \\(\frac{2}{8i}\\)
2. \\(\frac{3}{5i}\\)
3. \\(\frac{-5}{-5i}\\)
4. \\(\frac{-1}{-9i}\\)
5. \\(\frac{6}{-4i}\\)
6. \\(\frac{6 + 8i}{9i}\\)
7. \\(\frac{4 - 9i}{-6i}\\)
8. \\(\frac{-3 + 10i}{-6i}\\)
9. \\(\frac{-1 + 8i}{-i}\\)
10. \\(\frac{10 - 10i}{-5i}\\)
11. \\(\frac{5i}{-2 - 6i}\\)
12. \\(\frac{8i}{-1 + 3i}\\)

Explanation:

Step1: Simplify $\frac{2}{8i}$

To rationalize the denominator, multiply numerator and denominator by $i$ (since $i^2 = -1$).
$$\frac{2}{8i} \times \frac{i}{i} = \frac{2i}{8i^2}$$
Since $i^2 = -1$, substitute:
$$\frac{2i}{8(-1)} = \frac{2i}{-8} = -\frac{i}{4}$$

Step2: Simplify $\frac{3}{5i}$

Multiply numerator and denominator by $i$:
$$\frac{3}{5i} \times \frac{i}{i} = \frac{3i}{5i^2}$$
Substitute $i^2 = -1$:
$$\frac{3i}{5(-1)} = -\frac{3i}{5}$$

Step3: Simplify $\frac{-5}{-5i}$

Simplify the fraction first: $\frac{-5}{-5i} = \frac{1}{i}$
Multiply numerator and denominator by $i$:
$$\frac{1}{i} \times \frac{i}{i} = \frac{i}{i^2}$$
Substitute $i^2 = -1$:
$$\frac{i}{-1} = -i$$

Step4: Simplify $\frac{-1}{-9i}$

Simplify the fraction: $\frac{-1}{-9i} = \frac{1}{9i}$
Multiply numerator and denominator by $i$:
$$\frac{1}{9i} \times \frac{i}{i} = \frac{i}{9i^2}$$
Substitute $i^2 = -1$:
$$\frac{i}{9(-1)} = -\frac{i}{9}$$

Step5: Simplify $\frac{6}{-4i}$

Multiply numerator and denominator by $i$:
$$\frac{6}{-4i} \times \frac{i}{i} = \frac{6i}{-4i^2}$$
Substitute $i^2 = -1$:
$$\frac{6i}{-4(-1)} = \frac{6i}{4} = \frac{3i}{2}$$

Step6: Simplify $\frac{6 + 8i}{9i}$

Multiply numerator and denominator by $i$:
$$\frac{6 + 8i}{9i} \times \frac{i}{i} = \frac{(6 + 8i)i}{9i^2}$$
Expand the numerator: $6i + 8i^2$
Substitute $i^2 = -1$: $6i + 8(-1) = 6i - 8$
Denominator: $9(-1) = -9$
So, $\frac{6i - 8}{-9} = \frac{-8 + 6i}{-9} = \frac{8 - 6i}{9} = \frac{8}{9} - \frac{2i}{3}$

Step7: Simplify $\frac{4 - 9i}{-6i}$

Multiply numerator and denominator by $i$:
$$\frac{4 - 9i}{-6i} \times \frac{i}{i} = \frac{(4 - 9i)i}{-6i^2}$$
Expand numerator: $4i - 9i^2$
Substitute $i^2 = -1$: $4i - 9(-1) = 4i + 9$
Denominator: $-6(-1) = 6$
So, $\frac{9 + 4i}{6} = \frac{9}{6} + \frac{4i}{6} = \frac{3}{2} + \frac{2i}{3}$

Step8: Simplify $\frac{-3 + 10i}{-6i}$

Multiply numerator and denominator by $i$:
$$\frac{-3 + 10i}{-6i} \times \frac{i}{i} = \frac{(-3 + 10i)i}{-6i^2}$$
Expand numerator: $-3i + 10i^2$
Substitute $i^2 = -1$: $-3i + 10(-1) = -3i - 10$
Denominator: $-6(-1) = 6$
So, $\frac{-10 - 3i}{6} = -\frac{10}{6} - \frac{3i}{6} = -\frac{5}{3} - \frac{i}{2}$

Step9: Simplify $\frac{-1 + 8i}{-i}$

Multiply numerator and denominator by $i$:
$$\frac{-1 + 8i}{-i} \times \frac{i}{i} = \frac{(-1 + 8i)i}{-i^2}$$
Expand numerator: $-i + 8i^2$
Substitute $i^2 = -1$: $-i + 8(-1) = -i - 8$
Denominator: $-(-1) = 1$
So, $-8 - i$

Step10: Simplify $\frac{10 - 10i}{-5i}$

Simplify the fraction: $\frac{10 - 10i}{-5i} = \frac{2 - 2i}{-i}$
Multiply numerator and denominator by $i$:
$$\frac{2 - 2i}{-i} \times \frac{i}{i} = \frac{(2 - 2i)i}{-i^2}$$
Expand numerator: $2i - 2i^2$
Substitute $i^2 = -1$: $2i - 2(-1) = 2i + 2$
Denominator: $-(-1) = 1$
So, $2 + 2i$

Step11: Simplify $\frac{5i}{-2 - 6i}$

Multiply numerator and denominator by the conjugate of the denominator, which is $-2 + 6i$:
$$\frac{5i}{-2 - 6i} \times \frac{-2 + 6i}{-2 + 6i} = \frac{5i(-2 + 6i)}{(-2)^2 - (6i)^2}$$
Expand numerator: $-10i + 30i^2$
Substitute $i^2 = -1$: $-10i + 30(-1) = -10i - 30$
Denominator: $4 - 36i^2 = 4 - 36(-1) = 4 + 36 = 40$
So, $\frac{-30 - 10i}{40} = -\frac{3}{4} - \frac{i}{4}$

Step12: Simplify $\frac{8i}{-1 + 3i}$

Multiply numerator and denominator by the conjugate of the denominator, which is $-1 - 3i$:
$$\frac{8i}{-1 + 3i} \times \frac{-1 - 3i}{-1 - 3i} = \frac{8i(-1 - 3i)}{(-1)^2 - (3i)^2}$$
Expand numerator: $-8i - 24i^2$
Substitute $i^2 = -1$: $-8i - 24(-1) = -8i + 24$
Denominator: $1 - 9i^2 = 1 - 9(-1) = 1 + 9 = 10$
So, $\frac{24 - 8i}{10} = \frac{12}{5} - \frac{4i…

Answer:

1. $-\frac{i}{4}$
2. $-\frac{3i}{5}$
3. $-i$
4. $-\frac{i}{9}$
5. $\frac{3i}{2}$
6. $\frac{8}{9} - \frac{2i}{3}$
7. $\frac{3}{2} + \frac{2i}{3}$
8. $-\frac{5}{3} - \frac{i}{2}$
9. $-8 - i$
10. $2 + 2i$
11. $-\frac{3}{4} - \frac{i}{4}$
12. $\frac{12}{5} - \frac{4i}{5}$