Verified Answer :

  • For \(-6(3i)(-2i)\), simplify the product of coefficients to get \(36\), then multiply by \(i^2 = -1\) to get \(\boxed{-36}\).
  • For \(2(3-i)(-2+4i)\), expand the product \((3-i)(-2+4i)\) to \(-2+14i\), then distribute the 2 to obtain \(\boxed{-4+28i}\).

Explanation

  1. Problem Analysis

The problem requires simplifying two expressions involving the imaginary unit \(i\), where \(i^2 = -1\).
Expression 1 is \(-6(3i)(-2i)\).
Expression 2 is \(2(3-i)(-2+4i)\).
Our objective is to compute the final value of each expression in the standard complex form \(a + bi\).
We will use the property \(i^2 = -1\) and the distributive property of multiplication.

  1. Simplifying Expression 1: \(-6(3i)(-2i)\)

For the first expression, \(-6(3i)(-2i)\):

  1. Multiply the constant coefficients and the imaginary units: \(-6 \times 3 \times (-2) = 36\).
  2. Multiply the imaginary units: \(i \times i = i^2\).
  3. Since \(i^2 = -1\), the expression simplifies to \(36 \times (-1) = -36\).

The final result for the first expression is \(-36 + 0i\), or simply \(-36\).

  1. Simplifying Expression 2: \(2(3-i)(-2+4i)\)

For the second expression, \(2(3-i)(-2+4i)\):

  1. Expand the product of the binomials \((3-i)(-2+4i)\) using the distributive property:

\(3 \times (-2) = -6\)
\(3 \times 4i = 12i\)
\(-i \times (-2) = 2i\)
\(-i \times 4i = -4i^2 = -4(-1) = 4\)

  1. Combine the terms: \((-6 + 4) + (12i + 2i) = -2 + 14i\).
  2. Multiply the entire result by the external factor of 2:

\(2 \times (-2 + 14i) = -4 + 28i\).
The final result for the second expression is \(-4 + 28i\).

Examples

Complex numbers are used extensively in electrical engineering to represent alternating current (AC) circuits, where voltage and current are represented as phasors. Simplifying these expressions is equivalent to calculating circuit impedance or combining signals, which is fundamental for designing efficient electrical systems and signal processing filters. Understanding how to manipulate \(i\) ensures accurate calculation of these physical properties.

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