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Chapter 6: Problem 66

Write each expression in the form \(a+b i,\) where \(a\) and \(b\) are real numbers. $$ \frac{9-\sqrt{-18}}{-6} $$

Short Answer

Expert verified
-\frac{3}{2} + \frac{1}{2} i \(\sqrt{2}.\)

Step by step solution

01

Simplify the square root of a negative number

Rewrite \( \sqrt{-18} \) using the imaginary unit \( i \): \( \sqrt{-18} = \sqrt{-1 \cdot 18} = \sqrt{-1} \cdot \sqrt{18} = i \sqrt{18} \).
02

Simplify the square root of 18

Simplify \( \sqrt{18} \) as: \( \sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \sqrt{2} = 3 \sqrt{2} \). Therefore, \( \sqrt{-18} = 3 i \sqrt{2}.\)
03

Substitute back into the expression

Replace \( \sqrt{-18} \) in the original expression: \ \frac{9 - \sqrt{-18}}{-6} = \frac{9 - 3i \sqrt{2}}{-6}.\
04

Separate the fraction

Distribute the fraction \( \frac{9 - 3i \sqrt{2}}{-6} \) into two parts: \ \frac{9}{-6} - \frac{3i \sqrt{2}}{-6}.\
05

Simplify each term

Simplify each term separately: \ \frac{9}{-6} = -\frac{3}{2} \) and \( - \frac{3i \sqrt{2}}{-6} = -\frac{3}{6} i \sqrt{2} = \frac{1}{2} i \sqrt{2}.\
06

Combine the terms

Combine the simplified terms to write the expression in the form \(a + b i\): \ -\frac{3}{2} + \frac{1}{2} i \sqrt{2}. \

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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, denoted as \( i \), is a mathematical concept that allows us to work with the square roots of negative numbers. The imaginary unit is defined by the property \( i^2 = -1 \). This means whenever we have the square root of a negative number, we can rewrite it using \( i \). For example, \( \sqrt{-1} = i \). By extension, if we need to find the square root of any negative number, we can factor it into a product involving \( i \):
  • \( \sqrt{-18} = \sqrt{-1 \cdot 18} = \sqrt{-1} \cdot \sqrt{18} = i \sqrt{18} \)

This way, we transform the problem of dealing with a negative square root into a simpler problem involving the imaginary unit.
Simplifying Radicals
Simplifying radicals involves breaking down a radical expression into its simplest form. A radical expression is simplified when there are no more perfect square factors under the square root. Here鈥檚 how you simplify \( \sqrt{18} \):
  • Factorize 18: \( 18 = 9 \cdot 2 \)
  • Break it down: \( \sqrt{18} = \sqrt{9 \cdot 2} \)
  • Simplify: \( \sqrt{9 \cdot 2} = \sqrt{9} \sqrt{2} = 3 \sqrt{2} \)

By recognizing that 9 is a perfect square, we were able to pull out a whole number, simplifying the expression. For our original problem, this helped us convert \( \sqrt{-18} \) into \( 3i \sqrt{2} \).
Complex Number Form
A complex number is a number that has both a real part and an imaginary part. It is typically written in the form \( a + bi \), where \( a \) is the real part and \( b \) is the coefficient of the imaginary part. In our given problem, the goal was to rewrite the expression \( \frac{9 - \sqrt{-18}}{-6} \) in this complex number form. Here鈥檚 a summary of how we did it:
  • After simplifying \( \sqrt{-18} \) to \( 3i \sqrt{2} \), we replaced it back into the original expression: \( \frac{9 - 3i \sqrt{2}}{-6} \)
  • Separated the fraction: \( \frac{9}{-6} - \frac{3i \sqrt{2}}{-6} \)
  • Simplified each term: \( \frac{9}{-6} = -\frac{3}{2} \) and \( -\frac{3i \sqrt{2}}{-6} = \frac{1}{2} i \sqrt{2} \)
  • Combined the simplified terms: \( -\frac{3}{2} + \frac{1}{2} i \sqrt{2} \)

In this way, the final answer is a complex number in the form \( a + bi \), specifically \( -\frac{3}{2} + \frac{1}{2} i \sqrt{2} \).

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Most popular questions from this chapter

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