Rationalize Complex Number Worksheets (printable, online, answers ... - Free Printable

To simplify a fraction with complex numbers, we multiply the top (numerator) and bottom (denominator) by the conjugate of the denominator. The conjugate is found by changing the sign between the real and imaginary parts (e.g., the conjugate of $a + bi$ is $a - bi$). This removes the imaginary unit $i$ from the denominator because $(a+bi)(a-bi) = a^2 + b^2$.

Here are the step-by-step solutions for each problem:

1. $\frac{8 + 3i}{-3 + i}$

* Conjugate of denominator: $-3 - i$
* Multiply:
$$ \frac{(8 + 3i)(-3 - i)}{(-3 + i)(-3 - i)} $$
* Denominator: $(-3)^2 + (1)^2 = 9 + 1 = 10$
* Numerator: $8(-3) + 8(-i) + 3i(-3) + 3i(-i) = -24 - 8i - 9i - 3i^2$
Since $i^2 = -1$, this becomes $-24 - 17i + 3 = -21 - 17i$
* Result: $\frac{-21 - 17i}{10} = -\frac{21}{10} - \frac{17}{10}i$

2. $\frac{9 + 7i}{1 - 3i}$

* Conjugate of denominator: $1 + 3i$
* Multiply:
$$ \frac{(9 + 7i)(1 + 3i)}{(1 - 3i)(1 + 3i)} $$
* Denominator: $1^2 + 3^2 = 1 + 9 = 10$
* Numerator: $9(1) + 9(3i) + 7i(1) + 7i(3i) = 9 + 27i + 7i + 21i^2$
Becomes $9 + 34i - 21 = -12 + 34i$
* Result: $\frac{-12 + 34i}{10} = -\frac{12}{10} + \frac{34}{10}i = -\frac{6}{5} + \frac{17}{5}i$

3. $\frac{7 - 7i}{9 - i}$

* Conjugate of denominator: $9 + i$
* Multiply:
$$ \frac{(7 - 7i)(9 + i)}{(9 - i)(9 + i)} $$
* Denominator: $9^2 + 1^2 = 81 + 1 = 82$
* Numerator: $7(9) + 7(i) - 7i(9) - 7i(i) = 63 + 7i - 63i - 7i^2$
Becomes $63 - 56i + 7 = 70 - 56i$
* Result: $\frac{70 - 56i}{82}$. Divide top and bottom by 2: $\frac{35 - 28i}{41} = \frac{35}{41} - \frac{28}{41}i$

4. $\frac{5 + i}{8 + i}$

* Conjugate of denominator: $8 - i$
* Multiply:
$$ \frac{(5 + i)(8 - i)}{(8 + i)(8 - i)} $$
* Denominator: $8^2 + 1^2 = 64 + 1 = 65$
* Numerator: $5(8) - 5(i) + i(8) - i(i) = 40 - 5i + 8i - i^2$
Becomes $40 + 3i + 1 = 41 + 3i$
* Result: $\frac{41 + 3i}{65} = \frac{41}{65} + \frac{3}{65}i$

5. $\frac{-10 + i}{-1 + 3i}$

* Conjugate of denominator: $-1 - 3i$
* Multiply:
$$ \frac{(-10 + i)(-1 - 3i)}{(-1 + 3i)(-1 - 3i)} $$
* Denominator: $(-1)^2 + 3^2 = 1 + 9 = 10$
* Numerator: $(-10)(-1) + (-10)(-3i) + i(-1) + i(-3i) = 10 + 30i - i - 3i^2$
Becomes $10 + 29i + 3 = 13 + 29i$
* Result: $\frac{13 + 29i}{10} = \frac{13}{10} + \frac{29}{10}i$

6. $\frac{-3 - 5i}{-8 + 3i}$

* Conjugate of denominator: $-8 - 3i$
* Multiply:
$$ \frac{(-3 - 5i)(-8 - 3i)}{(-8 + 3i)(-8 - 3i)} $$
* Denominator: $(-8)^2 + 3^2 = 64 + 9 = 73$
* Numerator: $(-3)(-8) + (-3)(-3i) + (-5i)(-8) + (-5i)(-3i) = 24 + 9i + 40i + 15i^2$
Becomes $24 + 49i - 15 = 9 + 49i$
* Result: $\frac{9 + 49i}{73} = \frac{9}{73} + \frac{49}{73}i$

7. $\frac{1 + 2i}{5 + i}$

* Conjugate of denominator: $5 - i$
* Multiply:
$$ \frac{(1 + 2i)(5 - i)}{(5 + i)(5 - i)} $$
* Denominator: $5^2 + 1^2 = 25 + 1 = 26$
* Numerator: $1(5) - 1(i) + 2i(5) - 2i(i) = 5 - i + 10i - 2i^2$
Becomes $5 + 9i + 2 = 7 + 9i$
* Result: $\frac{7 + 9i}{26} = \frac{7}{26} + \frac{9}{26}i$

8. $\frac{3 + 3i}{9 - i}$

* Conjugate of denominator: $9 + i$
* Multiply:
$$ \frac{(3 + 3i)(9 + i)}{(9 - i)(9 + i)} $$
* Denominator: $9^2 + 1^2 = 81 + 1 = 82$
* Numerator: $3(9) + 3(i) + 3i(9) + 3i(i) = 27 + 3i + 27i + 3i^2$
Becomes $27 + 30i - 3 = 24 + 30i$
* Result: $\frac{24 + 30i}{82}$. Divide top and bottom by 2: $\frac{12 + 15i}{41} = \frac{12}{41} + \frac{15}{41}i$

Final Answer:
1. $-\frac{21}{10} - \frac{17}{10}i$
2. $-\frac{6}{5} + \frac{17}{5}i$
3. $\frac{35}{41} - \frac{28}{41}i$
4. $\frac{41}{65} + \frac{3}{65}i$
5. $\frac{13}{10} + \frac{29}{10}i$
6. $\frac{9}{73} + \frac{49}{73}i$
7. $\frac{7}{26} + \frac{9}{26}i$
8. $\frac{12}{41} + \frac{15}{41}i$