Simplifying Complex Expressions: (52i)(2+7i)2i(3i)^2
This article will guide you through simplifying the complex expression (52i)(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.
StepbyStep Simplification

Expand the first product:
(52i)(2+7i) = 5(2) + 5(7i)  2i(2)  2i(7i)
= 10 + 35i  4i  14i²

Simplify (3i)²:
(3i)² = 3² * i² = 9 * (1) = 9

Substitute the value of i² and simplify:
10 + 35i  4i  14i²  2i(9) = 10 + 35i  4i + 14 + 18i

Combine like terms:
(10 + 14) + (35  4 + 18)i = 24 + 49i
Conclusion
Therefore, the simplified form of the complex expression (52i)(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.