Question

Multiply. Write the answer in standard form. \(3 i(-5+i)\)

Short answer

After multiplying the complex numbers and writing the answer in standard form, the final answer is \(-3-15i\).

Step-by-step solution

Distribute

Distribute the 3i to each part of the second complex number. Multiply \(3i\) by \(-5\) to get \(-15i\) and \(3i\) by \(i\) to get \(3i^{2}\)

Simplify

Now simplify the expression. Remember that \(i^{2}\) is equal to -1. So, the expression becomes \(-15i-3\) .

Rewrite in Standard Form

The final step is to rewrite the simplified expression in standard form where real part comes first and then imaginary part. Here it will be \(-3-15i\)

Key concepts

Standard Form

Complex numbers often come in what we call "standard form," which makes them easy to read and understand. A complex number in standard form is generally written as \(a + bi\).
This format separates the real and imaginary parts, with "\(a\)" representing the real part and "\(bi\)" being the imaginary part.
For example, in the standard form \(-3 - 15i\), "\(-3\)" is the real component and "\(-15i\)" is the imaginary component.
It's standard practice to write the real part first, followed by the imaginary part.

Imaginary Unit

In the world of complex numbers, the imaginary unit is represented by the symbol \(i\).
This is no ordinary number. Instead, it's defined by a special property: \(i^2 = -1\).
That means \(i\) is the square root of \(-1\), something you can’t find on the real number line.
The imaginary unit allows us to solve equations that have solutions not possible with just real numbers. For instance, the expression \(3i(-5+i)\) involves using \(i\) to explore a whole new set of numbers, making math even more exciting!

Multiplication of Complex Numbers

Multiplying complex numbers may at first seem tricky, but it's very much like multiplying variables.
Let's consider the given problem, where you multiply a complex number by \(3i\) as in \(3i(-5+i)\).
Start by treating \(3i\) as a single unit and distribute it to both parts inside the parentheses:

• Multiply \(3i\) by \(-5\) to get \(-15i\).
• Multiply \(3i\) by \(i\), which gives you \(3i^2\).

Then you simplify using the property that \(i^2 = -1\), leading you to the final result in its most understandable form.

Distributive Property

The distributive property helps in multiplying expressions more effectively, and it plays a crucial role when dealing with complex numbers.
This property tells us that for any numbers or expressions \(a\), \(b\), and \(c\), the expression \(a(b + c)\) is equal to \(ab + ac\).
By applying the distributive property to the multiplication \(3i(-5+i)\), we ensure that \(3i\) multiplies with each term inside the parentheses. This results in:

• \(3i \cdot (-5) = -15i\)
• \(3i \cdot i\)

Thus, using the distributive property makes the task of multiplying a complex number by another term organized and straightforward.