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Chapter 9: Problem 1

Write each number as a multiple of \(i\). $$ \sqrt{-9} $$

Short Answer

Expert verified
\(\sqrt{-9} = 3i\)

Step by step solution

01

Understand the concept of imaginary numbers

The imaginary unit is represented by the symbol \(i\), where \(i\) is defined as \( \sqrt{-1} \). Any negative number under a square root can be expressed as a multiple of \(i\).
02

Simplify the square root

Consider \( \sqrt{-9} \). Recognize that \( -9 \) can be written as \( -1 \times 9 \). Hence, \( \sqrt{-9} = \sqrt{-1 \times 9} \).
03

Separate the square roots

Use the property of square roots \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). This gives us \( \sqrt{-9} = \sqrt{-1} \times \sqrt{9} \).
04

Express in terms of \(i\)

Since \( \sqrt{-1} = i \) and \( \sqrt{9} = 3 \), we can write \( \sqrt{-9} = i \times 3 \) or \(3i\).

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Key Concepts

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

square root
The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 * 3 = 9. When dealing with negative numbers under a square root, things get tricky. This is where imaginary numbers come into play. Imaginary numbers allow us to take the square root of negative numbers. To handle this, we use the imaginary unit, represented by the symbol i, which is defined as \( \sqrt{-1} \). This simplifies understanding complex numbers.

For any negative number under the square root, you can rewrite it to involve i. For example, \( \sqrt{-9} \) can be rewritten using i as a part of the solution.
negative numbers
Negative numbers are numbers less than zero. They represent a loss or absence of a quantity. In math, dealing with negative numbers during operations like square roots requires special handling. You cannot directly take the square root of a negative number using real numbers. Here鈥檚 where imaginary numbers come into play.

Let's look closer at \( \sqrt{-9} \). You can separate the negative sign from the number, writing -9 as \( -1 \times 9 \) . This makes it easier to handle the square root. By separating these, we use the property of square roots where \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). This lets us work with more familiar square roots.
multiple of i
The imaginary unit i helps in expressing results involving the square root of negative numbers. Since \( i = \sqrt{-1} \), you can rewrite square roots of negative numbers using i.

For example, \( \sqrt{-9} \) can be broken down as \( \sqrt{-1 \times 9} \). Using the property of square roots, we can separate it to \( \sqrt{-1} \times \sqrt{9} \). We know that \( \sqrt{-1} = i \) and \( \sqrt{9} = 3 \). So, \( \sqrt{-9} \) becomes \( i \times 3 \), which is written as \( 3i \).

You now have the number expressed as a multiple of i, making it simpler to work with complex numbers in mathematical problems.

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Most popular questions from this chapter

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