When faced with complex expressions involving imaginary numbers, it is essential to understand how to simplify them to their equivalent forms. One such expression is (4 + 7i)(3 + 4i), which involves multiplying two complex numbers. By utilizing the FOIL method and understanding the properties of imaginary numbers, we can determine the equivalent expression of this multiplication.
Simplifying the Expression (4 + 7i)(3 + 4i)
To simplify the expression (4 + 7i)(3 + 4i), we first distribute the terms using the distributive property. This involves multiplying each term in the first complex number by each term in the second complex number. By doing so, we get:
4(3) + 4(4i) + 7i(3) + 7i(4i)
This simplifies to:
12 + 16i + 21i + 28i^2
Utilizing the FOIL Method to Find the Equivalent Expression
After simplifying the expression, we can further simplify it by using the FOIL method. FOIL stands for First, Outer, Inner, Last, and it is a technique used to multiply two binomials. In this case, our binomials are (4 + 7i) and (3 + 4i). By applying the FOIL method, we get:
First: 4 3 = 12
Outer: 4 4i = 16i
Inner: 7i 3 = 21i
Last: 7i 4i = 28i^2
Combining these terms, we get the equivalent expression of (4 + 7i)(3 + 4i) as:
12 + 16i + 21i + 28i^2
In conclusion, determining the equivalent expression of (4 + 7i)(3 + 4i) may seem daunting at first glance, but by breaking it down step by step and utilizing methods such as the FOIL method, we can simplify complex expressions involving imaginary numbers. Understanding the properties and operations involving imaginary numbers is crucial in tackling such problems effectively. By following the systematic approach outlined above, we can confidently determine the equivalent expression of (4 + 7i)(3 + 4i) and enhance our understanding of complex number arithmetic.