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Chapter 8: Problem 8

In Exercises 1-12, write each expression as a complex number in standard form. If an expression simplifies to either a real number or a pure imaginary number, leave in that form. $$ \sqrt{-27} $$

Short Answer

Expert verified
\(\sqrt{-27} = 3\sqrt{3}i\)

Step by step solution

01

Identify Complex Numbers

The expression involves a square root of a negative number, specifically \(-27\). Remember that the square root of a negative number involves the use of 'i', the imaginary unit, where \(i = \sqrt{-1}\).
02

Simplify the Square Root of the Imaginary Unit

Since \(-27 = -1 \times 27\), we can write \(\sqrt{-27} = \sqrt{-1 \times 27} = \sqrt{-1} \times \sqrt{27}\). This becomes \(i \times \sqrt{27}\) because \(\sqrt{-1} = i\).
03

Simplify the Radical

Now, simplify \(\sqrt{27}\). We know \(27 = 9 \times 3\), and \(\sqrt{9} = 3\). Therefore, \(\sqrt{27} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3}\).
04

Express in Standard Form

Combine the results to express the number in standard form as a complex number. Substituting back, we have \(i \times 3\sqrt{3}\) or simply \(3\sqrt{3}i\).

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Key Concepts

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

Imaginary Unit
The imaginary unit, often denoted as
  • i, plays a fundamental role in the realm of complex numbers.
  • i is defined by the equation \(i = \sqrt{-1}\), which allows us to express the square roots of negative numbers.
  • With i, we can extend our number system beyond real numbers to include those numbers that were previously deemed 'impossible.'
Whenever encountering a square root of a negative number, simply extract i from the root, which signifies that we've stepped into the space of complex numbers.In other words, understanding i is key to grasping how square roots of negative numbers can yield valid, albeit imaginary, results.
Standard Form of Complex Numbers
A complex number is expressed in standard form as
  • \(a + bi\) where a and b are real numbers,
  • and i is the imaginary unit.
  • a is referred to as the real part, and
  • b is the imaginary part.
The standard form effectively separates what is 'real' from what is 'imaginary,' providing a clear and organized representation of complex numbers.This organization is crucial, especially when performing mathematical operations like addition, subtraction, or multiplication on complex numbers.For example, in the expression \(3\sqrt{3}i\), the real part would be 0, since it is a purely imaginary number.
Simplifying Radicals
When dealing with radicals, especially in the context of complex numbers, understanding simplification is essential.Radicals, or square roots, often need to be broken down into simpler, more manageable parts. Consider
  1. \(\sqrt{27}\),
  2. which can be expressed as \(\sqrt{9 \times 3}\).
  3. From here,
  4. we simplify to \(\sqrt{9} \times \sqrt{3} = 3\sqrt{3}\).
This step-by-step process of breaking down and simplifying helps not just in seeing the big picture but also in finding the simplest form of the expression.Mastering this technique is valuable for further mathematical applications.
Square Root of Negative Numbers
Taking the square root of a negative number is what initially introduces us to complex numbers. Typically,
  • the square root of a negative number doesn't have a real solution because you'd need a number that when squared results in a negative value.
  • Instead, i comes into play.
  • For example,
  • for \(\sqrt{-27}\), decomposing the expression into \(\sqrt{-1 \times 27}\) allows us to separate the components into \(\sqrt{-1} \times \sqrt{27}\).
  • The part \(\sqrt{-1}\) is equal to i.
This turns our negative square root into something tangible and manageable, using the imaginary unit.

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