Question

In Problems \(11-58,\) perform the indicated operation, and write each expression in the standard form \(a+b i .\) 48\. \(i^{7}+i^{5}+i^{3}+i\)

Short answer

0 + 0i

Step-by-step solution

Understand Powers of i

Recall that the imaginary unit i has cyclic powers: \[i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1\]These values repeat every four powers.

Simplify Each Term

Use the cyclic nature of i to simplify each term in the given expression:\[ i^7 = i^{4+3} = (i^4) \times (i^3) = 1 \times (-i) = -i \]\[ i^5 = i^{4+1} = (i^4) \times i = 1 \times i = i \]\[ i^3 = -i \]\[ i = i \]

Combine the Simplified Terms

Add the simplified terms together:\[ -i + i - i + i = 0 \]

Write in Standard Form

Write the result in the standard form \(a + bi\):\[ 0 + 0i \]

Key concepts

powers of imaginary unit

To grasp operations with complex numbers, it's crucial to understand the powers of the imaginary unit, denoted as \(i\). The imaginary unit \(i\) follows a specific cyclical pattern when raised to higher powers. This pattern repeats every four steps:

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

Knowing these values helps in simplifying complex expressions involving powers of \(i\). For example, \(i^7\) can be reduced using the fact that \(i^4 = 1\), thus: \[i^7 = i^{4+3} = (i^4) \times (i^3) = 1 \times (-i) = -i\]. Similarly, other powers can also be simplified efficiently by recognizing this cyclical behavior.

cyclic behavior

The cyclical nature of \(i\) simplifies complex number operations greatly. Since the powers of \(i\) repeat every four exponents, identifying patterns becomes easier. For example, \(i^5\) can be viewed as \(i^{4+1}\), where \(i^4 = 1\), thus: \[i^5 = (i^4) \times i = 1 \times i = i\].

\(i^6\) would then be \[i^6 = i^{4+2} = (i^4) \times (i^2) = 1 \times (-1) = -1\]. This cyclic behavior enables us to reduce higher powers of \(i\) by grouping them into sets of four: \(i^4k\) where \(k\) is any integer. This property is handy in simplifying expressions and solving mathematical problems involving complex numbers.

standard form of complex numbers

Complex numbers are commonly represented in the standard form, given by \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. The real part of the complex number is \(a\), and the imaginary part is \((bi)\).

When performing operations on complex numbers, ensuring the result is in the standard form is essential. For instance, in the exercise, after simplifying \(i^{7}+i^{5}+i^{3}+i\), we combine the terms to get \(-i + i - i + i = 0\), which can be written as \(0 + 0i\), representing a complex number with both real and imaginary parts equal to zero. This helps in clearly understanding and interpreting complex number expressions.

simplification of imaginary numbers

Simplifying terms involving the imaginary unit \(i\) is essential in handling complex number expressions efficiently. Start by breaking down each term using the cyclical behavior of \(i\). For the expression \(i^{7}+i^{5}+i^{3}+i\):

• Simplify \(i^7\): \[i^7 = -i\]
• Simplify \(i^5\): \[i^5 = i\]
• Simplify \(i^3\): \[i^3 = -i\]
• Simplify \(i\): \[i = i\]

Next, combine the simplified terms: \(-i + i - i + i = 0\). Always write the final result in standard form, which in this case is \(0 + 0i\). This process ensures the expression is easily interpretable and mathematically precise.