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Chapter 7: Problem 11

Express in terms of \(i\). $$ \sqrt{-5} $$

Short Answer

Expert verified
The expression simplifies to \( i \sqrt{5} \).

Step by step solution

01

- Recognize the imaginary unit

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

- Rewrite the square root as a product

Rewrite \( \sqrt{-5} \) as \( \sqrt{-1 \times 5} \).
03

- Apply the property of square roots

Use the property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \) to separate the terms: \( \sqrt{-1 \times 5} = \sqrt{-1} \times \sqrt{5} \).
04

- Substitute \( i \) for \( \sqrt{-1} \)

Replace \( \sqrt{-1} \) with \( i \): \( \sqrt{-1} \times \sqrt{5} = i \times \sqrt{5} \).
05

- Simplify the expression

The expression simplifies to \( i \sqrt{5} \).

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Key Concepts

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

Square Roots
Square roots are mathematical operations that are the opposite of squaring a number. If you square 3, you get 9. So, the square root of 9 is 3. The square root symbol is \( \sqrt{} \), so \( \sqrt{9} = 3 \). Square roots are used frequently in solving equations and understanding different properties of numbers. However, what happens when we have the square root of a negative number? That's where imaginary numbers come into play.
Normally, you can't get the square root of a negative number in the real world. There's no real number that gives you a negative when squared. Let's move to the imaginary unit to understand this better.
Imaginary Unit
To handle the square root of negative numbers, mathematicians invented the imaginary unit, denoted as \(i\). The imaginary unit is defined as \( i = \sqrt{-1} \). This simple definition allows us to work with square roots of negative numbers. For example, \( \sqrt{-4} = \sqrt{4 \times -1} = \sqrt{4} \times \sqrt{-1} \). Since \( \sqrt{4} = 2 \) and \( \sqrt{-1} = i \), we get \( \sqrt{-4} = 2i \).
This imaginary unit \(i\) forms the basis of complex numbers, where any number can be written as a combination of real and imaginary parts, such as \(a + bi\). Let's now understand the properties of radicals to see how they help in calculations involving imaginary numbers.
Properties of Radicals
Radicals, or square roots, have several important properties that make them useful in mathematics. One key property is:
  • \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \)
This property allows for the separation of the square root of a product into the product of the square roots. In our example for \( \sqrt{-5} \), we can write it as \( \sqrt{-1 \times 5} \) and then separate it into \( \sqrt{-1} \times \sqrt{5} \). Using the definition of the imaginary unit, \( \sqrt{-1} = i \), we can simplify the expression to \( i \times \sqrt{5} \).
These properties make it easier to work with complex and imaginary numbers, allowing us to manipulate and simplify expressions involving square roots of negative numbers efficiently.

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Most popular questions from this chapter

Let \(f(x)=x^{2} .\) Find each of the following. $$ f(\sqrt{2}+\sqrt{10}) $$

Find the midpoint of the segment with the given endpoints. $$ \left(-\frac{4}{5},-\frac{2}{3}\right) \text { and }\left(\frac{1}{8}, \frac{3}{4}\right) $$

Multiply. $$ (\sqrt{x+2}-\sqrt{x-2})^{2} $$

Simplify. $$ 7 x \sqrt{(x+y)^{3}}-5 x y \sqrt{x+y}-2 y \sqrt{(x+y)^{3}} $$

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. Write the answer using radical notation. $$ \frac{\sqrt[3]{(2 x+1)^{2}}}{\sqrt[5]{(2 x+1)^{2}}} $$
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