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Chapter 2: Problem 84

Simplify the complex number and write it in standard form. \((\sqrt{-2})^{6}\)

Short Answer

Expert verified
The complex number \((\sqrt{-2})^{6}\) in standard form is \(-4i\).

Step by step solution

01

Represent the Square Root of a Negative Number

Rewrite \(\sqrt{-2}\) as \( i \sqrt{2}\). This is done because the square root of a negative number (\(\sqrt{-1}\)) is represented by the imaginary unit \(i\).
02

Simplify Using the Exponential Rule

After rewriting \(\sqrt{-2}\) as \(i \sqrt{2}\), the task is then to simplify \((i \sqrt{2})^{6}\). Apply the rules of exponents, \(a^{mn} = (a^m)^n\), the expression can be written as \((i^6)(\sqrt{2}^6)\).
03

Simplify Further

Simplify the expression \((i^6)(\sqrt{2}^6)\).The result is \(-4i\). This is because \(i^6\) is equal to \(-i\), considering that \(i^4 = 1\), and \(\sqrt{2}^6\) equals to 8, hence the expression becomes \(-4i\).

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Key Concepts

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

Imaginary Unit
In the world of complex numbers, the imaginary unit is a crucial concept. This special number, denoted by the symbol \(i\), is defined as the square root of negative one, \(i = \sqrt{-1}\). This definition helps us work with square roots of negative numbers, which are not possible in the set of real numbers.
  • For any negative number \(-a\), the square root \(\sqrt{-a}\) can be expressed as \(i\sqrt{a}\).
  • This allows us to extend the real number system into the complex number system.
Understanding the imaginary unit is essential for simplifying complex expressions. When dealing with square roots of negative numbers, using \(i\) makes calculations more straightforward as seen in our exercise where \(\sqrt{-2}\) is rewritten as \(i\sqrt{2}\). With this transformation, we navigate seamlessly through otherwise unsolvable equations.
Exponentiation
Exponentiation involving complex numbers, especially with the imaginary unit, follows the same basic rules as real numbers. However, understanding how powers of \(i\) behave is key.
  • To exponentiate an expression like \(i^n\), it's important to grasp that \(i\) follows a cyclical pattern: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\), and then it repeats.
  • This cycle repeats every four exponents, making simplifying powers of \(i\) manageable.
In our example, the task involved simplifying \((i\sqrt{2})^6\). We break it into \((i^6)(\sqrt{2}^6)\) using the rule \(a^{mn} = (a^m)^n\). Recognizing that \(i^6\) simplifies to \(-i\), due to the cyclical property, helps us obtain the final answer effortlessly.
Standard Form of Complex Numbers
The standard form of a complex number expresses these numbers as a sum, involving both real and imaginary parts. It looks like \(a + bi\), where \(a\) is the real component and \(b\) is the imaginary component.
  • In the expression \(-4i\), the real part \(a\) is 0, and the imaginary part \(b\) is -4.
  • Writing complex numbers in this form ensures clarity and helps convey the distinct contributions from both parts.
In our example solution where we identified \(\sqrt{2}^6\) as 8, the complex expression ultimately simplified to \(-4i\). This expression is already in its standard form as \(0 - 4i\) given the zero real part.

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