Problem 19 - Precalculus Student Solutions Manual 5th

Question

Write each number as the product of a real number and i. $$\sqrt{-10}$$

Step-by-Step Solution

Answer

\(\root{-10} = \root{10} \cdot i\)

Step 1: Understand the Problem

We need to express the given number, \(\root{-10}\), as the product of a real number and the imaginary unit \(i\).

Step 2: Recall the Definition of i

The imaginary unit, \(i\), is defined as \(i = \root{-1}\). This helps us represent negative square roots.

Step 3: Rewrite the Expression

Rewrite \(\root{-10}\) using the definition of \(i\). \(\root{-10} = \root{10 \cdot -1}\).

Step 4: Split the Radicals

Use the property of radicals that \(\root{a \cdot b} = \root{a} \cdot \root{b}\). Therefore, we have \( \root{10 \cdot -1} = \root{10} \cdot \root{-1}\).

Step 5: Substitute i

Substitute \(i\) for \( \root{-1} \). This gives us \(\root{10} \cdot i\).

Key Concepts

square roots

When working with square roots, we're often asked to find a number which, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 multiplied by itself equals 9.
However, things become interesting when we deal with negative numbers. The square root of a negative number isn't a real number because no real number squared gives a negative result.
This is where imaginary numbers come into play. They allow us to work with square roots of negative numbers using the imaginary unit, denoted as \(i\).

complex numbers

Complex numbers are numbers that have two parts: a real part and an imaginary part. They are typically written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.
The term 'complex' might sound daunting, but essentially, such numbers extend our usual number system to handle the square roots of negative numbers.
For example, \(2 + 3i\) is a complex number where 2 is the real part and 3i is the imaginary part. This forms the basis of many higher mathematical concepts and provides tools to solve previously unsolvable problems.

imaginary unit

The imaginary unit, \(i\), is defined as the square root of -1. It's a fundamental concept in mathematics, particularly when dealing with complex numbers.
Here's how it works: since \(i^2 = -1\), we can use \(i\) to represent the square roots of negative numbers. For instance, \(\sqrt{-1} = i\). This definition enables us to break down expressions involving square roots of negatives into more manageable parts.
In our example, \(\sqrt{-10}\), we rewrite it as: \(\sqrt{10} \times \sqrt{-1}\). Using \(i\) for \(\sqrt{-1}\), we get \(\sqrt{10} \times i\). This elegantly shows how our original expression translates using the imaginary unit.