Question

1. $5(3 - 2i) + 3(1 + 4i)$

$5×3 - 5×2i + 3×1 + 3×4i$

1. $7(-3 + 4i) - 8(2 - 3i)$

$7×(-3) + 7×4i - 8×2 + 8×3i$

1. $-2(3 - 12i) + 5(-1 + 7i)$

$-2×3 + 2×12i + 5×(-1) + 5×7i$

1. $-(7 - 8i) - 2(3 - 4i)$

$-1×(7 - 8i) - 2×3 + 2×4i$

Explanation:

Problem 1: \( 5(3 - 2i) + 3(1 + 4i) \)

Step 1: Distribute the coefficients

Using the distributive property \( a(b + c)=ab + ac \), we get:
\( 5\times3 - 5\times2i+3\times1 + 3\times4i \)
\( = 15-10i + 3+12i \)

Step 2: Combine like terms

Combine the real parts (\( 15 + 3 \)) and the imaginary parts (\( - 10i+12i \)):
Real parts: \( 15 + 3=18 \)
Imaginary parts: \( -10i + 12i = 2i \)
So the result is \( 18 + 2i \)

Step 1: Distribute the coefficients

Using the distributive property \( a(b + c)=ab + ac \) and \( a(b - c)=ab - ac \), we get:
\( 7\times(-3)+7\times4i-8\times2+8\times3i \)
\(=-21 + 28i-16 + 24i \)

Step 2: Combine like terms

Combine the real parts (\( -21-16 \)) and the imaginary parts (\( 28i + 24i \)):
Real parts: \( -21-16=-37 \)
Imaginary parts: \( 28i+24i = 52i \)
So the result is \( -37 + 52i \)

Step 1: Distribute the coefficients

Using the distributive property \( a(b - c)=ab - ac \) and \( a(b + c)=ab + ac \), we get:
\( -2\times3+2\times12i+5\times(-1)+5\times7i \)
\(=-6 + 24i-5 + 35i \)

Step 2: Combine like terms

Combine the real parts (\( -6-5 \)) and the imaginary parts (\( 24i + 35i \)):
Real parts: \( -6-5=-11 \)
Imaginary parts: \( 24i + 35i=59i \)
So the result is \( -11 + 59i \)

Answer:

\( 18 + 2i \)

Problem 2: \( 7(-3 + 4i)-8(2 - 3i) \)