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Question 2 4 pts Simplify: 2(1+3i)-(2-4i)+(5-2i) 5+4i 13 9+6i 5+8i Question 3 4 pts Simplify (-9+sqrt(-81))/(3)

Question 22\newline44 pts\newlineSimplify:\newline2(1+3i)−(2−4i)+(5−2i) 2(1+3 i)-(2-4 i)+(5-2 i) \newline5+4i 5+4 i \newline1313\newline9+6i 9+6 i \newline5+8i 5+8 i \newlineQuestion 33\newline44 pts\newlineSimplify\newline−9+−813 \frac{-9+\sqrt{-81}}{3}

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Full solution

Q. Question 22\newline44 pts\newlineSimplify:\newline2(1+3i)−(2−4i)+(5−2i) 2(1+3 i)-(2-4 i)+(5-2 i) \newline5+4i 5+4 i \newline1313\newline9+6i 9+6 i \newline5+8i 5+8 i \newlineQuestion 33\newline44 pts\newlineSimplify\newline−9+−813 \frac{-9+\sqrt{-81}}{3}
  1. Question prompt: Question prompt: Simplify the complex number expression 2(1+3i)−(2−4i)+(5−2i)2(1+3i)-(2-4i)+(5-2i).
  2. Distribute and combine: First, distribute the 22 into the first set of parentheses: 2×1+2×3i=2+6i2 \times 1 + 2 \times 3i = 2 + 6i.
  3. Combine real parts: Combine the distributed result with the other terms: (2+6i)−(2−4i)+(5−2i)(2 + 6i) - (2 - 4i) + (5 - 2i).
  4. Combine imaginary parts: Now, simplify by combining like terms. First, combine the real parts: 2−2+5=52 - 2 + 5 = 5.
  5. Final simplified form: Next, combine the imaginary parts: 6i+4i−2i=8i6i + 4i - 2i = 8i.
  6. Question prompt: Combine the real and imaginary parts to get the final simplified form: 5+8i5 + 8i.
  7. Recognize −81\sqrt{-81}: Question prompt: Simplify the expression −9+−813\frac{-9+\sqrt{-81}}{3}.
  8. Calculate 81\sqrt{81}: Recognize that −81\sqrt{-81} is the square root of a negative number, which can be expressed as 81⋅i\sqrt{81} \cdot i, since ii is the imaginary unit where i2=−1i^2 = -1.
  9. Substitute in expression: Calculate the square root of 8181, which is 99. So, −81=9i\sqrt{-81} = 9i.
  10. Divide by 33: Substitute 9i9i for −81\sqrt{-81} in the expression: (−9+9i)/(3)(-9+9i)/(3).
  11. Simplify terms: Divide both terms in the numerator by 33: (−9/3)+(9i/3)(-9/3) + (9i/3).
  12. Simplify terms: Divide both terms in the numerator by 33: (−9/3)+(9i/3)(-9/3) + (9i/3).Simplify each term: −3+3i-3 + 3i.

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