## Multiplying Complex Numbers: (6 + 7i)(6 - 7i)

This article explores the multiplication of complex numbers, specifically the product of (6 + 7i) and (6 - 7i).

### Understanding Complex Numbers

Complex numbers are numbers 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 (i² = -1).

### Multiplying Complex Numbers

Multiplying complex numbers is similar to multiplying binomials, using the distributive property (or FOIL method).

**Step 1: Expand the product:**

(6 + 7i)(6 - 7i) = 6(6 - 7i) + 7i(6 - 7i)

**Step 2: Distribute:**

= 36 - 42i + 42i - 49i²

**Step 3: Simplify using i² = -1:**

= 36 - 49(-1)

**Step 4: Combine real terms:**

= 36 + 49

**Step 5: Final result:**

= **85**

### Key Observation

Notice that the result of multiplying (6 + 7i) and (6 - 7i) is a real number (85). This is because (6 - 7i) is the complex conjugate of (6 + 7i).

**The complex conjugate of a complex number a + bi is a - bi**. Multiplying a complex number by its conjugate always results in a real number. This property is often used to simplify expressions involving complex numbers.

### Conclusion

The product of (6 + 7i) and (6 - 7i) is 85. This demonstrates the concept of multiplying complex numbers and the special case of multiplying a complex number by its conjugate, which always yields a real number.