Question

Write the expression as a complex number in standard form. \(2 i^3(5-12 i)\)

Short answer

The expression \(2 i^3(5-12 i)\) simplifies to \(-24 - 10i\) in standard form.

Step-by-step solution

Substitute value

The property of \(i\) needs to be remembered, where \(i^3 = -i\). So substitute \(i^3\) with \(-i\) in the given expression. This gives us \(2(-i)(5-12i)\).

Distribute the multiplication

Distribute \(2(-i)\) within the brackets, which gives us \(-10i + 24i^2\).

Simplify further

Here, represent \(i^2\) as -1 (since \(i^2 = -1\)). This gives us \(-10i + 24(-1)\) which simplifies down to \(-10i - 24\).

Write in standard form

The standard form of a complex number is \(a + bi\) so rearrange \(-10i - 24\) in this form to get \(-24 - 10i\).

Key concepts

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental component in the world of complex numbers. It satisfies the equation \(i^2 = -1\). This property is the cornerstone for defining complex numbers. Unlike real numbers, imaginary numbers provide solutions for equations that wouldn't otherwise have real solutions. For example, the equation \(x^2 + 1 = 0\) doesn't have a real number solution, but it does have solutions \(x = i\) and \(x = -i\).
This makes the imaginary unit essential in various mathematical fields, including engineering and physics. When dealing with powers of \(i\), follow these basic rules:

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)

After \(i^4\), the cycle repeats, making these rules very convenient when simplifying expressions with higher powers of \(i\). For example, in the problem \(2i^3(5-12i)\), knowing that \(i^3 = -i\) helps simplify the expression considerably.

Standard Form

The standard form of a complex number is expressed as \(a + bi\), where \(a\) and \(b\) are real numbers. In this form, \(a\) is the real part, and \(b\) is the imaginary part. For the complex number to be properly written in standard form, it should be arranged such that the real part comes before the imaginary part. This makes it easier to identify and manipulate both components separately.
In the given exercise, the expression is transformed into \(-24 - 10i\), where \(-24\) is the real part, and \(-10i\) is the imaginary part. To achieve this form:

• Start from any other organization of terms.
• Perform arithmetic operations, ensuring that \(i^2 = -1\) is applied when necessary.
• Rearrange terms to fit the \(a + bi\) format.

This structured approach helps in solving or further analyzing complex number operations.

Complex Number Operations

Operations involving complex numbers, such as addition, subtraction, multiplication, and division, follow specific rules that incorporate both their real and imaginary parts. Mastery of these operations is essential for handling complex numbers in various applications.
In multiplication, such as the scenario in the exercise with \(2(-i)(5-12i)\), distribute the terms to combine real and imaginary parts. Keep the following tips in mind:

• When multiplying, expand using the distributive property.
• Simplify by substituting \(i^2\) with \(-1\).
• Combine like terms, ensuring real and imaginary parts are clear.

In addition, when numbers involve powers of \(i\), substitution such as \(i^2 = -1\) helps further simplify expressions. For this problem, after distribution, combining terms provides a clear path to the final answer of \(-24 - 10i\). By following these steps and understanding the properties of \(i\), complex number operations become much simpler to handle.