(-4i)(-5-5i)-7(1-4i)

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Expressions

This article will guide you through simplifying the complex expression (-4i)(-5-5i)-7(1-4i).

Understanding Complex Numbers

Before we dive into the simplification, let's briefly revisit what complex numbers are. Complex numbers are numbers that can be expressed in the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit, defined as the square root of -1.

Simplifying the Expression

Let's break down the simplification process step by step:

1. Distribute:

   • First, we distribute -4i across the parentheses in the first term: (-4i)(-5-5i) = 20i + 20i²
   • Next, we distribute -7 across the parentheses in the second term: -7(1-4i) = -7 + 28i

2. Substitute i² = -1:

   • Remember that i² = -1. Substitute this value into the expression: 20i + 20i² = 20i + 20(-1) = -20 + 20i

3. Combine Like Terms:

   • Now we have: (-20 + 20i) + (-7 + 28i)
   • Combining the real terms (-20 and -7) and the imaginary terms (20i and 28i): -20 - 7 + 20i + 28i = -27 + 48i

Final Result

Therefore, the simplified form of the complex expression (-4i)(-5-5i)-7(1-4i) is -27 + 48i.