Question

What is the value of \((2+8 i)(1-4 i)-(3-2 i)(6+4 i)\) ? (Note: \(i=\sqrt{-1}\) ) A) 8 B) 26 C) 34 D) 50

Short answer

The value of the expression is 8 (Option A).

Step-by-step solution

Multiply the first two complex numbers

To multiply the first two complex numbers, \((2+8i)(1-4i)\), we apply the distributive property (also known as the FOIL method for binomials): \((2+8i)(1-4i) = 2(1) + 2(-4i) + 8i(1) + 8i(-4i)\)

Simplify the first part

Now we simplify the expression we got in the previous step: \(2(1) + 2(-4i) + 8i(1) + 8i(-4i) = 2 - 8i + 8i - 32i^2\) We know that \(i^2 = -1\), so we substitute this into our expression: \(2 - 8i + 8i - 32(-1) = 2 + 32 = 34\)

Multiply the second two complex numbers

To multiply the second two complex numbers, \((3-2i)(6+4i)\), we again apply the distributive property: \((3-2i)(6+4i) = 3(6) + 3(4i) - 2i(6) - 2i(4i)\)

Simplify the second part

Now we simplify the expression we got in the previous step: \(3(6) + 3(4i) - 2i(6) - 2i(4i) = 18 + 12i - 12i - 8i^2\) Again, we substitute \(i^2\) with \(-1\): \(18 + 12i - 12i - 8(-1) = 18 + 8 = 26\)

Subtract the second part from the first part

Now that we have simplified both parts, we can subtract the second part from the first part: \(34 - 26 = 8\) So the value of the given expression is 8, which corresponds to option A.