Multiply and simplify. 133) 5-6+7i 134)8i(4 3i) 135) 3i-7 + 12) 137) (4 -6)(3 -6i) Identify the conjugate of each complex number, then multiply 136) (3-4)(2 + i) 38) (1-2, 3 2 (5 3 2 3 the number and its conjugate. 139) 2 7i Divide. Write the quotient in the form a + bi. 141) 5 140) 2 3i 142) -3i 2+5i 143) 144) 9-4i 6- i 145) Simplify. 147) 0 149) 3 146) -2 +6i 148)1 150) i24
The Correct Answer and Explanation is :
Let’s go through each problem step-by-step:
133) (5 – 6 + 7i)
This expression is simply a combination of real numbers and imaginary numbers. You can combine the real terms:
[
5 – 6 = -1
]
So, the result is:
[
-1 + 7i
]
134) (8i(4 + 3i))
To multiply a complex number by a real number, distribute the multiplication across the terms inside the parentheses:
[
8i \times 4 = 32i
]
[
8i \times 3i = 24i^2
]
Since (i^2 = -1), we can substitute that in:
[
24i^2 = 24(-1) = -24
]
Thus, the result is:
[
32i – 24
]
135) (3i – 7 + 12)
Here, simplify by combining real and imaginary terms:
[
-7 + 12 = 5
]
So the result is:
[
5 + 3i
]
137) ((4 – 6)(3 – 6i))
To multiply these complex numbers, apply the distributive property (FOIL):
[
(4)(3) = 12
]
[
(4)(-6i) = -24i
]
[
(-6)(3) = -18
]
[
(-6)(-6i) = 36i
]
Now, combine the results:
[
12 – 24i – 18 + 36i
]
[
(12 – 18) + (-24i + 36i) = -6 + 12i
]
Thus, the result is:
[
-6 + 12i
]
136) ((3 – 4i)(2 + i))
We use FOIL to expand this:
[
(3)(2) = 6
]
[
(3)(i) = 3i
]
[
(-4i)(2) = -8i
]
[
(-4i)(i) = -4i^2 = 4 \quad (\text{since } i^2 = -1)
]
Now, combine the terms:
[
6 + 3i – 8i + 4
]
[
(6 + 4) + (3i – 8i) = 10 – 5i
]
Thus, the result is:
[
10 – 5i
]
138) ((1 – 2i)(3 – 2i))
Use FOIL to expand:
[
(1)(3) = 3
]
[
(1)(-2i) = -2i
]
[
(-2i)(3) = -6i
]
[
(-2i)(-2i) = 4i^2 = -4
]
Now, combine the terms:
[
3 – 2i – 6i – 4
]
[
(3 – 4) + (-2i – 6i) = -1 – 8i
]
Thus, the result is:
[
-1 – 8i
]
139) (2 + 7i)
The conjugate of a complex number (a + bi) is (a – bi). So, the conjugate of (2 + 7i) is:
[
2 – 7i
]
140) (\frac{5}{2 + 3i})
To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator:
The conjugate of (2 + 3i) is (2 – 3i). So we multiply both the numerator and denominator by (2 – 3i):
[
\frac{5}{2 + 3i} \times \frac{2 – 3i}{2 – 3i} = \frac{5(2 – 3i)}{(2 + 3i)(2 – 3i)}
]
First, simplify the denominator:
[
(2 + 3i)(2 – 3i) = 2^2 – (3i)^2 = 4 – (-9) = 4 + 9 = 13
]
Now, simplify the numerator:
[
5(2 – 3i) = 10 – 15i
]
Thus, the result is:
[
\frac{10 – 15i}{13} = \frac{10}{13} – \frac{15}{13}i
]
141) (\frac{-3i}{2 + 5i})
Multiply the numerator and denominator by the conjugate of the denominator:
The conjugate of (2 + 5i) is (2 – 5i). Multiply both the numerator and denominator by (2 – 5i):
[
\frac{-3i}{2 + 5i} \times \frac{2 – 5i}{2 – 5i} = \frac{-3i(2 – 5i)}{(2 + 5i)(2 – 5i)}
]
Simplify the denominator:
[
(2 + 5i)(2 – 5i) = 2^2 – (5i)^2 = 4 – (-25) = 4 + 25 = 29
]
Now, simplify the numerator:
[
-3i(2 – 5i) = -6i + 15i^2 = -6i – 15 = -15 – 6i
]
Thus, the result is:
[
\frac{-15 – 6i}{29} = \frac{-15}{29} – \frac{6}{29}i
]
142) (\frac{9 – 4i}{6 – i})
Multiply both the numerator and denominator by the conjugate of the denominator:
The conjugate of (6 – i) is (6 + i). Multiply both the numerator and denominator by (6 + i):
[
\frac{9 – 4i}{6 – i} \times \frac{6 + i}{6 + i} = \frac{(9 – 4i)(6 + i)}{(6 – i)(6 + i)}
]
Simplify the denominator:
[
(6 – i)(6 + i) = 6^2 – i^2 = 36 – (-1) = 36 + 1 = 37
]
Now, simplify the numerator using distributive property (FOIL):
[
(9)(6) = 54
]
[
(9)(i) = 9i
]
[
(-4i)(6) = -24i
]
[
(-4i)(i) = -4i^2 = 4
]
Now, combine the terms:
[
54 + 9i – 24i + 4 = 58 – 15i
]
Thus, the result is:
[
\frac{58 – 15i}{37} = \frac{58}{37} – \frac{15}{37}i
]
Final Answers:
- ( -1 + 7i )
- ( 32i – 24 )
- ( 5 + 3i )
- ( -6 + 12i )
- ( 10 – 5i )
- ( -1 – 8i )
- Conjugate of ( 2 + 7i ): ( 2 – 7i )
- (\frac{10}{13} – \frac{15}{13}i)
- (\frac{-15}{29} – \frac{6}{29}i)
- (\frac{58}{37} – \frac{15}{37}i)
The process of simplifying and multiplying/dividing complex numbers relies on applying the distributive property (FOIL) and using the fact that (i^2 = -1). For division, we multiply the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part in the denominator and express the result in the form (a + bi).