Problem 49 - College Algebra

Question

For the following exercises, evaluate the expressions, writing the result as a simplified complex number. $$ i^{7}\left(1+i^{2}\right) $$

Step-by-Step Solution

Answer

The simplified complex number is 0.

Step 1: Simplify the Powers of i

Recall that the imaginary unit i satisfies \( i^2 = -1 \). This allows us to further determine other powers of i: \( i^3 = i imes i^2 = i imes (-1) = -i \) and \( i^4 = i^2 imes i^2 = (-1) imes (-1) = 1 \). Since powers of i cycle every four, \( i^7 = i^{4+3} = (i^4)^1 imes i^3 = 1 imes (-i) = -i \).

Step 2: Simplify the Parenthesis

Inside the parenthesis, we have \( 1 + i^2 \). Since \( i^2 = -1 \), we can substitute to get:\(1 + i^2 = 1 - 1 = 0.\)

Step 3: Multiply the Simplified Terms

The expression now is \( i^7 imes (1 + i^2) = -i imes 0 \). Since anything multiplied by 0 is 0, the result of the entire expression is: \(0.\)

Step 4: Finalize the Answer

After performing all operations, the simplified complex number of the given expression is: 0, because any imaginary or real number multiplied by zero results in zero.

Key Concepts

Imaginary Unit

The concept of the imaginary unit, represented by the symbol \(i\), is fundamental in the field of complex numbers. In mathematics, the imaginary unit is defined as the square root of -1. This means \(i^2 = -1\). Understanding this property is crucial for simplifying complex number expressions.

The imaginary unit is used to form complex numbers, which are numbers in the form \(a + bi\), where \(a\) and \(b\) are real numbers. The part \(bi\) contains the imaginary unit, making it the 'imaginary part' of the complex number. The real and imaginary parts can be combined in calculations, using their respective operations.

Remember, while \(i\) itself is not a real number, the expression \(ai\) or \(bi\) can be manipulated using algebraic rules just like real numbers.

Powers of i

When working with complex numbers, understanding the behavior of powers of \(i\) is essential. The powers of \(i\) follow a cyclical pattern every four exponent values:

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

After \(i^4\), the cycle repeats itself with \(i^5 = i\), \(i^6 = -1\), and so on. Knowing this cycle allows us to simplify expressions with higher powers of \(i\). To determine a power such as \(i^7\), divide the exponent by 4 and use the remainder to find the equivalent power from the cycle. For example, since 7 divided by 4 leaves a remainder of 3, \(i^7 = i^3 = -i\).

This cyclical property greatly simplifies calculations by reducing higher powers to one of the four basic forms.

Simplified Complex Expression

Simplifying complex number expressions involves performing operations using both the real and imaginary parts, sometimes needing a few helpful steps like recognizing powers of \(i\) and substituting known values. In our example exercise, simplifying the expression \(i^7(1 + i^2)\) demonstrates this.

First, use the cyclical property of \(i\) to reduce \(i^7\) to \(-i\). Next, handle the parenthesis: substitute \(i^2 = -1\), simplifying \(1 + i^2\) to 0. Multiplying further, \(-i \times 0\) results in 0.

This shows how a seemingly complex expression can be turned into a very simple result through understanding and applying properties of complex numbers. The expression's final form is thus a clear indicator of how powerful these mathematical tools are, enabling students to simplify and solve problems with ease.