## Multiplying Complex Numbers: A Look at (9+2i)(9-2i)

This article will explore the multiplication of the complex numbers (9+2i) and (9-2i), illustrating the concept of complex conjugates and their significance in simplifying calculations.

### Understanding Complex Numbers

Complex numbers are numbers that consist of a real part and an imaginary part. The imaginary part is a multiple of the imaginary unit *i*, where *i* is defined as the square root of -1.

For example, the complex number (9+2i) has a real part of 9 and an imaginary part of 2i.

### Complex Conjugates

The complex conjugate of a complex number is formed by changing the sign of its imaginary part. In our case, the complex conjugate of (9+2i) is (9-2i).

The product of a complex number and its conjugate always results in a real number. This property is crucial in simplifying calculations involving complex numbers.

### Multiplication of (9+2i) and (9-2i)

To multiply these two complex numbers, we can apply the distributive property or the FOIL method (First, Outer, Inner, Last).

Let's use the FOIL method:

**First:**9 * 9 = 81**Outer:**9 * -2i = -18i**Inner:**2i * 9 = 18i**Last:**2i * -2i = -4i²

Adding these terms together:

81 - 18i + 18i - 4i²

Since i² is defined as -1, we can substitute:

81 - 4(-1) = 81 + 4 = 85

### Conclusion

The product of (9+2i) and (9-2i) is **85**, a real number. This demonstrates the property of complex conjugates, where multiplying a complex number by its conjugate eliminates the imaginary component, leaving only a real number.

This understanding is essential for various mathematical applications, including solving equations, simplifying expressions, and working with complex functions.