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When studying complex numbers, the imaginary unit is a foundational concept. It is represented by the symbol \( i \), and is defined as \( i^2 = -1 \). This definition is crucial because it allows us to understand and manipulate numbers that go beyond the real number line.
This might seem abstract at first, but \( i \) effectively extends our number system. For example, when you perform operations with complex numbers like \( 2 + i \), you use the property \( i^2 = -1 \) to help simplify expressions.
Using \( i \), any complex number can be expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers. Remember, real numbers are just a subset of complex numbers where \( b = 0 \). This introduction of the imaginary unit \( i \) enables us to broaden our mathematical toolkit drastically.

Multiplying complex numbers can initially seem daunting, but it can be tackled using a structured approach.
Just like multiplying binomials in algebra, you can use the distributive property to handle the multiplication of complex numbers. For the complex expression \((2+i)^3\), you'll start by multiplying two \((2+i)\) terms to simplify the expression. This involves expanding \((2+i)\) by itself to get \((2+i)(2+i) = 4 + 2i + 2i + i^2\).
After calculating \(i^2 = -1\), you will simplify the expression further to \(3 + 4i\), and then multiply by \(2 + i\) again. Using the distributive property once more, you get \((3+4i)(2+i) = 6 + 3i + 8i + 4i^2\). After a similar simplification process, this results in \(2 + 11i\).

• Step 1: Expand the expression, using FOIL (First, Outer, Inner, Last).
• Step 2: Apply \(i^2 = -1\) to simplify.
• Step 3: Combine like terms.

Mastering these steps helps in simplifying such expressions efficiently.

Once a complex expression is expanded and all terms are collected, simplification is the next step. Simplifying ensures that the final expression is in its simplest form, often expressed as \( a + bi \) where \( a \) and \( b \) are real numbers.
Simplification involves two main parts: combining like terms and using the definition of the imaginary unit. For example, in solving \((2+i)^3\), you began by expanding and applying \( i^2 = -1 \) during multiplication. The expression \(6 + 3i + 8i + 4i^2\) becomes simplified further by organizing and calculating to reach \(2 + 11i\).
This simplification is achieved by following these key steps:

• Group real and imaginary components of the expanded expression together.
• Apply \(i^2 = -1\) and simplify the imaginary number calculations.
• Write the final result in the form \( a + bi \).

These steps ensure that the expression is clear, precise, and in the standard form, making any further calculations or comparisons more straightforward.