Question

1. (5 + 6i) + (2 - 3i) - (4 - 2i) - simplify.

a) 3 + i
b) 3 + 11i
c) 3 + 5i
d) 3 + 7i

1. (3 + 4i) + (7 - 2i) - (5 + 3i) simplify.

a) 5 - 3i
b) 5 + i
c) 5 + 3i
d) 5 - i

1. find the equivalent exponential form for 2 log₄ x = -1.

a) 4⁻¹ = ½x²
b) 16⁻¹ = x²
c) 4⁻² = x
d) 4⁻¹ = ½x

1. find the equivalent exponential form for 3 log₅ x = -1.

a) 5⁻¹ = 3x
b) 5⁻¹ = x²
c) 5⁻³ = x
d) 5⁻¹ = ⅓x³

1. simplify (-2 + 6i)².

a) -32 - 24i
b) 4 + 36i
c) 4 - 12i
d) -32 + 24i

1. simplify (3 - 4i)².

a) 25 + 24i
b) 25 - 24i
c) -7 + 24i
d) -7 - 24i

Explanation:

Step1: Correct typo in Problem 1

Assume the problem is $(5 + 6i) + (2 - 3i) - (4 - 2i)$ (correcting "4-21" to "4-2i").

Step2: Combine real parts

$5 + 2 - 4 = 3$

Step3: Combine imaginary parts

$6i - 3i + 2i = 5i$

Step4: Simplify Problem 2

$(3 + 4i) + (7 - 2i) - (5 + 3i)$ (correcting "d+3i" to "5+3i"): real parts $3+7-5=5$, imaginary parts $4i-2i-3i=-i$

Step5: Rewrite Problem 3 using log rules

$2\log_4 x = -1 \Rightarrow \log_4 x^2 = -1 \Rightarrow 4^{-1} = x^2$

Step6: Rewrite Problem 4 using log rules

$3\log_5 x = -1 \Rightarrow \log_5 x^3 = -1 \Rightarrow 5^{-1} = x^3$ (correcting "log6" to "log5")

Step7: Expand Problem 5

$(-2 + 6i)^2 = (-2)^2 + 2(-2)(6i) + (6i)^2 = 4 - 24i - 36 = -32 - 24i$ (matching option a/d typo)

Step8: Expand Problem 6

$(3 - 4i)^2 = 3^2 - 2(3)(4i) + (4i)^2 = 9 - 24i - 16 = -7 - 24i$ (correcting options if needed)

Answer:

1. c) 3 + 5i
2. d) 5 - i
3. a) $4^{-1} = x^2$
4. d) $5^{-1} = x^3$
5. a) -32 - 24i
6. d) -7 - 24i