Problem 36 Find the indicated powers of com... [FREE SOLUTION]

Chapter 7: Problem 36

Find the indicated powers of complex numbers. $$(-9 i)^{2}$$

Short Answer

Expert verified

-81

Step by step solution

Understanding the problem

To find $ (-9i)^{2} $, need to raise the complex number $ -9i $ to the power of 2. Remember that $ i $ is the imaginary unit where $ i^{2} = -1 $.

Apply the exponent

Raise both the real and imaginary parts to the power of 2: $ (-9i)^{2} = (-9)^2 \times (i)^2 $.

Calculate separately

First calculate $ (-9)^2 = 81 $, then $ (i)^2 = -1 $. Thus, $ (-9i)^{2} = 81 \times -1 $.

Combine the results

Combine the results from the previous step: $ 81 \times -1 = -81 $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Number Powers

Raising complex numbers to a power might seem complicated at first, but with the right understanding, it becomes manageable. Here, we shall focus on raising the complex number $-9i$ to the power of 2. This process includes:
* Understanding the exponential form of complex numbers
* Separately calculating the powers of the real and imaginary parts

Key point to remember: When raising a complex number to a power, you can separately raise its magnitude and imaginary unit part. For instance, in our problem, $(-9i)^2$, you can compute the power of -9 independently of the power of $i$.

Combining these results, we find $(-9)^2 $ gives us 81 and $i^2$ results in -1.

Imaginary Unit

The imaginary unit, denoted as $i$, is fundamental in complex number calculations. It's defined as $i $ where:
* $i^2 = -1$
* Maths rules involving \i\ remain essential in complex number manipulation

In our solution example, remember that when we square the imaginary unit part $i$, it always results in -1. So, knowing that $(-9i)^2 = (-9)^2 \times (i)^2 $, we calculate $(-9)^2 $, yielding 81, and then $i^2 $, which gives -1. We multiply these results to get the final outcome of -81.

Staying mindful of such imaginary unit rules significantly simplifies the complex number operations.

Exponent Rules

Exponent rules simplify the process of raising numbers to a power, and the same applies to complex numbers. Here are some foundational rules to remember:
* $a^m \times a^n = a^{m+n}$
* $(a^m)^n = a^{m \times n}$
* Applied to imaginary units, keep in mind \i^2 = -1, i^3 = -i, \ and \i^4 = 1\

In the solution $(-9i)^2 $, observe that the rule $ (a \times b)^m = a^m \times b^m $ helps us break down the problem efficiently. We apply $-9i \times -9i = (-9)^2 \times i^2$, leading to \81 \times -1 \ resulting in -81.

Mastering these rules helps in both simplifying and accurately computing powers of complex numbers.

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91影视

Find the indicated powers of complex numbers. $$(-9 i)^{2}$$

Short Answer

Step by step solution

Understanding the problem

Apply the exponent

Calculate separately

Combine the results

Key Concepts

Complex Number Powers

Imaginary Unit

Exponent Rules

One App. One Place for Learning.

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