## Simplifying Complex Numbers: (5 + 3i)(5 - 3i)

In mathematics, complex numbers are 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.

This article will walk through simplifying the expression **(5 + 3i)(5 - 3i)** into standard form.

### Understanding Complex Numbers

**Real Part:**The real part of a complex number is the term without the imaginary unit*i*. In the expression (5 + 3i), the real part is 5.**Imaginary Part:**The imaginary part of a complex number is the term multiplied by the imaginary unit*i*. In the expression (5 + 3i), the imaginary part is 3.

### Simplifying the Expression

We can simplify the expression (5 + 3i)(5 - 3i) by applying the distributive property (also known as FOIL).

**1. Distribute the terms:**
(5 + 3i)(5 - 3i) = (5 * 5) + (5 * -3i) + (3i * 5) + (3i * -3i)

**2. Simplify the multiplication:**
= 25 - 15i + 15i - 9i²

**3. Remember that i² = -1:**
= 25 - 15i + 15i - 9(-1)

**4. Combine like terms:**
= 25 + 9

**5. Final Answer:**
= **34**

### Conclusion

By applying the distributive property and substituting i² with -1, we have successfully simplified the expression (5 + 3i)(5 - 3i) to its standard form, **34**. This result demonstrates an important property of complex numbers: multiplying a complex number by its conjugate (the complex number with the opposite sign for the imaginary part) results in a real number. In this case, the conjugate of (5 + 3i) is (5 - 3i).