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Imaginary numbers might sound intimidating, but they are simpler than you might think! An imaginary number is a number that can be written as a real number multiplied by the imaginary unit, which is represented by the symbol \(i\). The key property of \(i\) is that \(i^2 = -1\). This property is what defines the imaginary unit and helps us work out operations involving imaginary numbers.
When dealing with square roots of negative numbers like \(\sqrt{-15}\), we express them using \(i\). Specifically, the expression \(\sqrt{-15}\) can be broken down as \(\sqrt{-1 \times 15} = \sqrt{-1} \times \sqrt{15} = i\sqrt{15}\). This conversion helps us to easily deal with negative square roots and simplifies further calculations in complex number operations.

• Imaginary Unit: \(i^2 = -1\)
• Examples: \(i\sqrt{15}, i\sqrt{9} = 3i\)
• Application: Useful for handling roots of negative numbers

The standard form of a complex number is a concise way of expressing numbers that have both real and imaginary parts. It is usually written in the form \(a + bi\), where \(a\) is the real component, and \(bi\) is the imaginary component.
In our exercise with \((\sqrt{-15})^2\), the standard form becomes apparent after calculation. Initially, we have \(\sqrt{-15} = i\sqrt{15}\). Upon squaring, we find that \((i\sqrt{15})^2\) results in \(-15\). There is no imaginary part (as it is \(+ 0i\)), and hence the standard form is \(-15 + 0i\). This form helps in identifying and separating the real and imaginary parts, which is vital for complex number operations.

• Standard Form: \(a + bi\)
• Real component: \(a\)
• Imaginary component: \(bi\)

Operations involving complex numbers are just as essential as operations with real numbers. They include addition, subtraction, multiplication, and division. Let's take a closer look at how multiplication works with imaginary numbers.
In the exercise, \((i\sqrt{15})^2\) is a multiplication operation. When multiplying two imaginary numbers, such as \(i\sqrt{15}\) by itself, we follow the distribution property. First, multiply the imaginary units, which gives us \(i^2 = -1\). Then, multiply the real numbers inside the square root \(\sqrt{15} \times \sqrt{15} = 15\). So, \((i\sqrt{15})^2 = i^2 \times 15 = -1 \times 15 = -15\).
These operations can simplify the process of working with complex numbers, allowing us to find results efficiently and in a structured manner.

• Multiplication: Distribute and apply \(i^2 = -1\)
• Addition/Subtraction: Combine like terms (real with real, imaginary with imaginary)
• Example Calculation: \((i\sqrt{15})^2 = -15\)