Question

Perform the indicated operation, and write each expression in the standard form \(a+\) bi. $$ i^{7}\left(1+i^{2}\right) $$

Short answer

0

Step-by-step solution

Simplify the Inner Expression

First, simplify the expression inside the parentheses: \(1 + i^2\).Since \(i^2 = -1\), we get:\[ 1 + i^2 = 1 + (-1) = 0 \]

Multiply the Result with the Outer Expression

Now, multiply the simplified inner expression by the outer expression: \(i^7 (0)\).Any number multiplied by 0 is 0, so:\[ i^7 \times 0 = 0 \]

Confirm the Standard Form

The standard form for a complex number is \(a + bi\).Here, the expression simplifies to 0, which can be written as:\(0 + 0i\).

Key concepts

Imaginary Unit

The imaginary unit, represented as \( i \), is a fundamental element used in complex numbers. It is defined by the property that \( i^2 = -1 \). This definition helps bridge the gap between real numbers and imaginary numbers. Imaginary numbers are those that can be written in the form \( bi \), where \( b \) is a real number, and they are essential for performing operations and solving equations that real numbers alone can't handle.
Here are some important properties of the imaginary unit:

• \( i^3 = i^2 \times i = -i \)
• \( i^4 = (i^2)^2 = 1 \)
• \( i^5 = i^4 \times i = i \)

Notice that every power of \( i \) cycles through these values: 1, \( i \), -1, and -\( i \). This cyclical pattern is fundamental when simplifying expressions involving powers of \( i \).

Standard Form

The standard form of a complex number is \(a + bi \), where:

• \( a \) is the real part
• \( b \) is the imaginary part

The process of converting a complex number into this form involves simplifying any expressions involving \( i \) and ensuring there are no powers of \( i \) higher than one. For example, consider the original problem \( i^7 (1 + i^2) \):
First, we simplify the expression inside the parentheses:
Since \( i^2 = -1 \), we have:\[ 1 + i^2 = 1 + (-1) = 0 \]
Next, we multiply this result by \( i^7 \):
\[ i^7 \times 0 = 0 \]
Since the result is 0, it can be written in the standard form as:
\[ 0 + 0i \]
This confirms that the complex number is completely simplified and follows the standard form.

Operations with Complex Numbers

When working with complex numbers, you will often perform various operations such as addition, subtraction, multiplication, and division. Understanding these operations is crucial for handling problems involving complex numbers.
In the provided exercise--simplifying \( i^7 (1 + i^2) \)--we break it down as follows:

• First, simplify \( 1 + i^2 \). Since \( i^2 = -1 \), we get \( 1 + (-1) = 0 \).
• Next, multiply \( i^7 \) by 0. Any number multiplied by 0 equals 0, so \( i^7 \times 0 = 0 \).

Multiplication and other operations with complex numbers often require following specific steps to ensure accuracy. Always handle the imaginary unit according to its defined properties, and you'll find that complex numbers can be managed just as easily as real numbers.