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Chapter 1: Problem 18

Write each expression in terms of \(i\) and simplify. (See Examples \(1-2)\) $$\sqrt{-100}$$

Short Answer

Expert verified
The expression \(\sqrt{-100}\) simplifies to \(10i\).

Step by step solution

01

Recognize the Imaginary Unit

The imaginary unit is denoted by the symbol \(i\), and is defined as \(i = \sqrt{-1}\).
02

Express the Radical in Terms of \(i\)

Rewrite the expression \(\sqrt{-100}\) by factoring out \(\sqrt{-1}\). This will give \(\sqrt{-100} = \sqrt{100} \cdot \sqrt{-1}\).
03

Simplify the Square Roots

Simplify each part of the expression separately. We know that: \(\sqrt{100} = 10\) and \(\sqrt{-1} = i\). Thus, \(\sqrt{100} \cdot \sqrt{-1} = 10i\).
04

Write the Final Simplified Expression

Combining the results from the previous step, we get the simplified form of \(\sqrt{-100}\) which is \(10i\).

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Key Concepts

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

complex numbers
Complex numbers are numbers that have both a real part and an imaginary part. They are written in the form: \(a + bi\) where \(a\) is the real part and \(bi\) is the imaginary part. The imaginary unit \(i\) is defined as \(\sqrt{-1}\).
Complex numbers encompass all real numbers (when \(b = 0\)) and all purely imaginary numbers (when \(a = 0\)). Understanding complex numbers is essential for solving many mathematical problems, including simplifying square roots of negative numbers.
square roots
The square root of a number \(x\) is a number that, when multiplied by itself, returns \(x\). Square roots are typically denoted as \(\sqrt{x}\). When working with real numbers, \(\sqrt{x}\) is only defined for non-negative \(x\). However, the concept of square roots extends to negative numbers by introducing the imaginary unit \(i\).
In the case of negative numbers, such as \(-100\), we use \(i\) to express the square root. For example, \(\sqrt{-100}\ = \sqrt{100 \cdot (-1)}\ = \sqrt{100} \cdot \sqrt{-1}\). This allows us to simplify \(\sqrt{-100}\) into a form involving \(i\).
imaginary unit i
The imaginary unit \(i\) is a mathematical concept defined as \(\sqrt{-1}\). It allows us to work with the square roots of negative numbers, which are not possible in the realm of real numbers. Here are some key points to remember:
  • \(i^2 = -1\)
  • \(i^3 = -i\)
  • \(i^4 = 1\)
To simplify square roots involving negative numbers, we factor out \(\sqrt{-1}\) as \(i\). For example, \(\sqrt{-100}\ = \sqrt{100 \cdot (-1)} = \sqrt{100} \cdot \sqrt{-1} = 10i\).
simplification steps
Simplifying expressions involving imaginary numbers requires following a few straightforward steps. Let's go through the example \(\sqrt{-100}\):
  • Step 1: Recognize the Imaginary Unit. Remember that \(i = \sqrt{-1}\).
  • Step 2: Express the Radical in Terms of \(i\). Rewrite the expression \(\sqrt{-100}\) by factoring out \(\sqrt{-1}\). This gives us \(\sqrt{-100} = \sqrt{100} \cdot \sqrt{-1}\).
  • Step 3: Simplify the Square Roots. Determine the square root of the positive part and the \(\sqrt{-1}\): \(\sqrt{100} = 10\) and \(\sqrt{-1} = i\).
  • Step 4: Write the Final Simplified Expression. Combining the simplified parts, we get \(10i\).
Following these steps allows you to handle more complex expressions involving square roots and imaginary numbers.

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Most popular questions from this chapter

A subway ride is \(\$ 2.25\) per ride. a. Write a model for the \(\operatorname{cost} C\) (in \$) for \(x\) rides on the subway. b. A commuter can purchase an unlimited-ride MetroCard for \(\$ 89\) per month. How many rides are required for a commuter to save money by buying the MetroCard?

The lengths of the sides of a triangle are given by three consecutive integers greater than \(1 .\) What are the possible values for the shortest side if the perimeter is not to exceed \(24 \mathrm{ft}\) ?

Solve the inequality, and write the solution set in interval notation. \(\left|\frac{1}{2} p-6\right| \geq 0.01\)

Write a three-part inequality to represent the given statement. A tennis player must play in the "open" division of a tennis tournament if the player's age \(a\) is over 18 yr and under 25 yr.

Discuss the difference between the products \((a+b)(a-b)\) and \((a+b i)(a-b i)\).
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