The Value of the Product (2i)(3i)

When dealing with complex numbers, the product of two imaginary numbers can be a fascinating concept to explore. In this article, we will dive into the step-by-step process of finding the value of the product (2i)(3i).

Understanding Complex Numbers and Imaginary Units

Complex numbers are a combination of real and imaginary parts, where the imaginary part is represented by the imaginary unit, i. The imaginary unit i is defined as the square root of -1, meaning i² = -1.

Multiplying Complex Numbers

To find the product of two complex numbers, we can use the distributive property. In this case, we have the product (2i)(3i).

Let's break down the steps:

1. Multiply the real parts: 2 × 3 = 6
2. Multiply the imaginary parts: i × i = i²
3. Simplify the imaginary part: i² = -1

Putting it all together, we get:

(2i)(3i) = 6i² (2i)(3i) = 6(-1) (2i)(3i) = -6

Therefore, the value of the product (2i)(3i) is -6.

Properties of Imaginary Numbers

It's important to note that when multiplying imaginary numbers, the following properties hold:

1. i² = -1
2. i³ = -i
3. i⁴ = 1

These properties can be used to simplify and manipulate expressions involving imaginary numbers.

Conclusion

In this article, we have explored the value of the product (2i)(3i) by applying the properties of complex numbers and imaginary units. By understanding the steps involved, you can confidently work with similar expressions involving the multiplication of imaginary numbers.