Question

Perform the indicated operation, and write each expression in the standard form \(a+\) bi. $$ (3 i)^{4}+1 $$

Short answer

82 + 0i

Step-by-step solution

Understand the problem

The problem requires performing the operation with complex numbers and writing the result in the standard form, which is in the form of \(a + bi\).

Expand the exponentiation

Calculate \((3i)^4\). Start by recognizing that \((3i)^4\) means multiplying \(3i\) by itself four times.

Compute \( (3i)^2 \)

First, calculate \((3i)^2\). This gives:\[ (3i)^2 = 3i \times 3i = 9i^2 \]Since we know that \(i^2 = -1\), we have:\[ 9i^2 = 9 \times (-1) = -9 \]

Compute \( (3i)^4 \)

Next, square \(-9\) to find \((3i)^4\):\[ (3i)^4 = (-9)^2 = 81 \]

Add the constant term

Now add the constant term \(+1\) to the result from Step 4:\[ (3i)^4 + 1 = 81 + 1 = 82 \]

Write in standard form

Since there is no imaginary part to the final answer, write the result in the standard form \(a + bi\):\[ 82 = 82 + 0i \]

Key concepts

Standard Form

When working with complex numbers, the standard form is crucial. This form is expressed as \(a + bi\) where \(a\) is the real part, and \(b\) is the imaginary part. The term \(bi\) involves \(i\), which represents the imaginary unit. In our example, the initial task requires taking \((3i)^4 + 1\) and rewriting it in this standard form. We calculated the value to be 82 with no imaginary component. Hence, the result in standard form is simply \(82 + 0i\).

To understand better, here's a quick summary:

• \(a\) and \(b\) must be real numbers.


• \(i\) is the imaginary unit equivalent to \(\sqrt{-1}\).


• Any complex number without an explicit imaginary part has an imaginary part equal to zero.

Exponentiation of Complex Numbers

Exponentiation of a complex number involves raising the number to a given power. In our example, we raised \(3i\) to the 4th power, \((3i)^4\). To break it down:

• First, compute \((3i)^2\) to get \(9i^2\).


• Since \(i^2 = -1\), \(9i^2\) translates to \(9(-1) = -9\).


• Next, square the result to find \((-9)^2 = 81\).


• Add the constant term \(+1\) to get the final result \(82\) in standard form.

Exponentiation of complex numbers can follow basic algebra rules, but it requires careful handling due to the properties of \(i\).

Always simplify intermediate steps using known properties of the imaginary unit to prevent mistakes.

Imaginary Unit Properties

The imaginary unit, denoted by \(i\), is defined by the property that \(i^2 = -1\). Recognizing and using this property is essential in complex number arithmetic. For instance, in our exercise:

• We dealt with \(i\) first by calculating \((3i)^2 = 9i^2\).


• Recognizing \(i^2 = -1\) allows us to simplify to \(-9\).


• Applying this property simplifies higher powers of \(i\) as in \((3i)^4\).

Additional properties include:

• \(i^3 = i \cdot i^2 = i(-1) = -i\)


• \(i^4 = (i^2)^2 = (-1)^2 = 1\)

Understanding these can aid in breaking down and simplifying operations involving complex numbers. Always refer back to these key properties during calculations to maintain accuracy and clarity.