Problem 19 - Intermediate Algebra

Question

Express each number in terms of \(i\). $$ \sqrt{-7} $$

Step-by-Step Solution

Answer

The solution is \(i\sqrt{7}\).

Step 1: Understanding Complex Numbers

Complex numbers are written in the form \(a + bi\), where \(i\) is the imaginary unit with the property \(i^2 = -1\). Our task is to express \(\sqrt{-7}\) in terms of \(i\).

Step 2: Rewrite the Radicand with \(-1\)

The number under the square root, called the radicand, is \(-7\). This can be rewritten as \((-1) \times 7\). Thus, \(\sqrt{-7} = \sqrt{-1 \times 7}\).

Step 3: Separate the Roots

Using the property of square roots, \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\), we can separate \(\sqrt{-1 \times 7}\) as \(\sqrt{-1} \times \sqrt{7}\).

Step 4: Replace \(\sqrt{-1}\) with \(i\)

By definition, \(\sqrt{-1} = i\). Therefore, our expression becomes \(i \times \sqrt{7}\). This is the number expressed in terms of \(i\).

Key Concepts

Imaginary Unit

Complex numbers utilize the imaginary unit, denoted as \(i\), to extend the number system into a new dimension. The defining characteristic of \(i\) is its fundamental property: \(i^2 = -1\). This unique property allows us to simplify expressions involving square roots of negative numbers. By definition, the square root of \(-1\) is represented by \(i\).
When dealing with complex numbers or expressions like \(\sqrt{-7}\), this property of \(i\) becomes incredibly useful. Instead of struggling with the concept of taking the square root of a negative number—a task impossible in the realm of real numbers—we transition into the complex realm using \(i\), providing a structured way to handle such operations easily and consistently.

Square Roots

Square roots are mathematical operations that "undo" squaring. When you take the square root of a number \(x\), you are seeking a number \(y\) such that \(y^2 = x\).
For positive numbers, square roots are straightforward. However, negative numbers pose a unique challenge because there's no real number \(y\) for which \(y^2\) is negative.
In these situations, we turn to complex numbers. For example, to express \(\sqrt{-7}\) in terms of \(i\), we use the property that any negative number under the square root can be written as the product of \(-1\) and a positive number. Thus, \(\sqrt{-7} = \sqrt{-1 \times 7}\).
This allows us to separate the radicals into \(\sqrt{-1} \times \sqrt{7}\), and we can substitute \(\sqrt{-1}\) with \(i\), ultimately simplifying to \(i \times \sqrt{7}\). This highlights how square roots connect with the imaginary unit to effectively express complex numbers.

Expressing Complex Numbers

Expressing numbers in terms of \(i\) is commonplace when dealing with complex numbers. A complex number is typically written as \(a + bi\), where \(a\) and \(b\) are real numbers.
The term \(a\) represents the real part, and \(bi\) is the imaginary part. Whenever you take the square root of a negative number, such as \(-7\), knowing how to express it in this form is invaluable.
To do so, identify the negative under the square root and rewrite it by factoring out the \(-1\): \(\sqrt{-7} = \sqrt{-1 \times 7}\). Then break it down using this property: \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\).
This allows you to express it as \(i \times \sqrt{7}\), which is now in an understandable form that conforms with the complex number format. Doing this makes it easier to work with, whether you're adding, subtracting, or performing any operations on complex numbers.