2.jpeg "(6 + 7i)2")
Expanding (6 + 7i)<sup>2</sup>
In this article, we will explore the process of expanding the complex number (6 + 7i)<sup>2</sup>.
Understanding Complex Numbers
A complex number is a number 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.
Expanding the Expression
To expand (6 + 7i)<sup>2</sup>, we can use the distributive property or the FOIL method:
(6 + 7i)<sup>2</sup> = (6 + 7i)(6 + 7i)
Using FOIL:
- First: 6 * 6 = 36
- Outer: 6 * 7i = 42i
- Inner: 7i * 6 = 42i
- Last: 7i * 7i = 49i<sup>2</sup>
Combining the terms:
36 + 42i + 42i + 49i<sup>2</sup>
Since i<sup>2</sup> = -1, we can simplify:
36 + 42i + 42i - 49 = -13 + 84i
Therefore, (6 + 7i)<sup>2</sup> expands to -13 + 84i.
Conclusion
We have successfully expanded the complex number (6 + 7i)<sup>2</sup> to -13 + 84i. This demonstrates the process of squaring a complex number and the importance of understanding the properties of the imaginary unit i.