Problem 6 - Algebra 2 and Trigonometry

Question

In \(3-18,\) write each number in terms of \(i\) $$ -\sqrt{-36} $$

Step-by-Step Solution

Answer

The expression simplifies to \\(-6i\\).

Step 1: Identify the Negative Argument

The problem requires us to express the square root of a negative number, \(-\sqrt{-36}\). First, identify that you are dealing with \(-36\), which is negative.

Step 2: Rewrite the Square Root of a Negative Number

Rewrite \(-36\) in terms of \(i\), where \(i = \sqrt{-1}\). So, \(-36 = 36 \times (-1)\), which means we can rewrite it as \(\sqrt{36} \times \sqrt{-1}\).

Step 3: Simplify Using the Value of i

Simplify \(\sqrt{36} \times \sqrt{-1}\) to become \(6 \times i\) since \(\sqrt{36} = 6\) and \(\sqrt{-1} = i\).

Step 4: Apply the Negative Sign Outside the Square Root

Apply the negative sign from the original expression \(-\sqrt{-36}\), giving us \(-6i\).

Key Concepts

Imaginary Unit

In mathematics, there's a special component known as the **imaginary unit**, denoted by the symbol \(i\). It is a fundamental idea when dealing with complex numbers. The imaginary unit is defined as the square root of negative one, meaning \(i = \sqrt{-1}\).

Understanding the concept of the imaginary unit helps to simplify expressions involving the square roots of negative numbers. Since the square root of any negative number can't be represented on the real number line, the introduction of \(i\) allows us to explore new numbers known as imaginary numbers, which are an extension of the real numbers. To put it more simply, any time you see \(\sqrt{-1}\), you can replace it with \(i\).

With this tool, we can simplify and manipulate mathematical expressions that involve the square roots of negative numbers, as seen in complex equations and certain fields like engineering and physics.

Square Roots

Square roots are a way of representing a number that, when multiplied by itself, gives the original number. The expression \(\sqrt{x}\) seeks a number that when squared equals \(x\).

For instance, \(\sqrt{36} = 6\) because \(6 \times 6 = 36\). When dealing with square roots on negative numbers, the concept shifts slightly because a real number squared always results in a positive number. This is where the imaginary unit \(i\) becomes really handy as it helps handle these cases elegantly.

For example, when working through an expression like \(\sqrt{-36}\), we can rewrite this as \(\sqrt{36 \times -1}\). This turns into \(\sqrt{36} \times \sqrt{-1}\), which simplifies down to \(6i\). The imaginary unit \(i\) thus becomes a bridge that allows handling square roots of negatives efficiently.

Negative Numbers

Negative numbers are below zero and take on an important role in many mathematical calculations. They are represented with a minus sign \(-\) before the number. In dealing with square roots involving these numbers, additional steps are necessary. Normally, finding a square root of a negative directly wouldn’t be possible within the realm of real numbers.

However, by understanding the interplay between negative numbers and the imaginary unit \(i\), we can navigate through this effortlessly. In expressions like \(-\sqrt{-36}\), you start by factoring the number into components like \(36 \times -1\). This allows for extracting the square root of the positive component as seen with \(\sqrt{36} = 6\), and acknowledging the square root of \(-1\) as \(i\). Bringing these together gives \(6i\), while the leading negative sign means the final expression is \(-6i\).

This intertwining of negative numbers and the imaginary unit is key in solving and simplifying problems involving negative square roots.