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The standard form of a complex number is key to understanding how these numbers are represented and manipulated. A complex number is typically written as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. In this form, both components are clearly displayed, making it easier to see what each part contributes to the complex number.
When asked to express a final result in standard form, ensure every term fits into this structure. This clarity is vital for performing further operations or solving equations involving complex numbers.
For example, in the exercise, the final expression \(10 + 18i\) is in standard form, with \(10\) as the real part and \(18i\) as the imaginary part.

In a complex number, the real part is the component without the imaginary unit \(i\). Often represented by \(a\) in standard form \(a + bi\), the real part stands on its own value.
The real part itself can be any real number, positive or negative. It is important to correctly identify and combine these parts in a complex expression.

• In our example, the real numbers from the original expression are \(22, -5,\) and \(-7\).
• When combined, they form \(22 - 5 - 7\).

Correctly adding these gives you the overall real part, which is \(10\). This is straightforward, but understanding it is foundational in working with complex numbers.

The imaginary part of a complex number involves the unit \(i\), providing a unique twist to our numbers. In the standard form \(a + bi\), \(bi\) indicates the imaginary part.
Here, the values multiply by \(i\), denoting the imaginary unit, which has the property \(i^2 = -1\).

• Looking back at our exercise, you'll find the imaginary terms as \(8i\) and \(10i\).
• To simplify, add these imaginary numbers: \(8i + 10i\).

This results in a combined imaginary part of \(18i\). Understanding how to handle these terms is crucial when simplifying or solving complex expressions.

Combining like terms is an essential algebraic skill, especially crucial when working with complex numbers. This involves grouping and simplifying terms with the same variable or unit.
Complex numbers have both real and imaginary terms, which must be combined separately.

• Our example breaks down as follows:
• Real parts: \(22, -5, -7\). This simplifies to \(10\).
• Imaginary parts: \(8i, 10i\). This combines to \(18i\).

Once like terms are combined, you arrive at the expression \(10 + 18i\). This step is vital in ensuring any complex operation is performed accurately, allowing the equation to be solved methodically.