(5-2i)(2+7i)-2i(3i)^2

Jasonbradley

2 min read

·

Jun 16, 2024

21

Share

Simplifying Complex Expressions: (5-2i)(2+7i)-2i(3i)^2

This article will guide you through simplifying the complex expression (5-2i)(2+7i)-2i(3i)^2. We'll use the distributive property, the rules of exponents, and the fundamental properties of complex numbers to achieve this.

Understanding Complex Numbers

Before we start, let's briefly review some essential concepts about complex numbers:

• Complex numbers are numbers of the form a + bi, where a and b are real numbers, and i is the imaginary unit.
• Imaginary unit (i) is defined as the square root of -1, meaning i² = -1.
• Distributive property applies to complex numbers just as it does to real numbers.
• Multiplication of complex numbers follows the same rules as multiplication of binomials.

Step-by-Step Simplification

1. Expand the first product:

   (5-2i)(2+7i) = 5(2) + 5(7i) - 2i(2) - 2i(7i)

   = 10 + 35i - 4i - 14i²

2. Simplify (3i)²:

   (3i)² = 3² * i² = 9 * (-1) = -9

3. Substitute the value of i² and simplify:

   10 + 35i - 4i - 14i² - 2i(-9) = 10 + 35i - 4i + 14 + 18i

4. Combine like terms:

   (10 + 14) + (35 - 4 + 18)i = 24 + 49i

Conclusion

Therefore, the simplified form of the complex expression (5-2i)(2+7i)-2i(3i)² is 24 + 49i. This result is a complex number in standard form (a + bi), where 24 is the real part and 49 is the imaginary part.