Problem 22 - Intermediate Algebra

Question

Express each number in terms of \(i\). $$ \sqrt{-28} $$

Step-by-Step Solution

Answer

\(\sqrt{-28} = 2i\sqrt{7}\)

Step 1: Understand the square root of a negative number

The number inside the square root, -28, is negative. To address this, realize that the square root of a negative number can be expressed in terms of the imaginary unit \(i\), where \(i = \sqrt{-1}\).

Step 2: Separate the negative sign

Re-write \(\sqrt{-28}\) as \(\sqrt{-1 \times 28}\). This takes advantage of the property \(\sqrt{ab} = \sqrt{a} \times \sqrt{b}\).

Step 3: Apply the imaginary unit

Use the property \(\sqrt{-1} = i\). Therefore, \(\sqrt{-1 \times 28} = \sqrt{-1} \times \sqrt{28} = i \times \sqrt{28}\).

Step 4: Simplify the square root of 28

The number 28 can be factored into 4 and 7, where 4 is a perfect square. This allows us to simplify as follows: \(\sqrt{28} = \sqrt{4 \times 7} = \sqrt{4} \times \sqrt{7} = 2\sqrt{7}\).

Step 5: Combine terms

Combine the results from previous steps: \(i \times 2\sqrt{7}\). Simplifying, we express \(\sqrt{-28}\) in terms of \(i\) as \(2i\sqrt{7}\).

Key Concepts

Understanding Square Roots

The square root of a number is essentially a value that, when multiplied by itself, gives the original number. For positive numbers, this is straightforward: the square root of 9 is 3, because 3 times 3 equals 9. But what happens when we're dealing with negative numbers? This brings us to imaginary numbers and specifically, the number \(i\). Imaginary numbers come into play because no real number multiplied by itself equals a negative number. This is where \(i = \sqrt{-1}\) becomes crucial. By defining this imaginary unit, we can determine the square roots of negative numbers by expressing them in terms of \(i\).
To compute something like \(sqrt{-28}\), recognize that this is the same as \(\sqrt{-1 \times 28}\), which can be rewritten using \(\sqrt{a} \times \sqrt{b}\) into \(\sqrt{-1} \times \sqrt{28}\). The \(\sqrt{-1}\) is replaced with \(i\), allowing us to handle negative square roots easily.

Exploring Negative Numbers

Negative numbers appear on the left side of the number line. They are less than zero and often cause confusion, especially when operations like square roots come into play. When considering square roots, negative numbers are tricky because the result isn't a number you're used to seeing. Typically, squaring any real number results in a positive value, but if we need to work with a square root of a negative number, we need to think outside the typical realm.
This is where imagining numbers, especially the use of \(i\), becomes essential. By understanding that \(i\) is \(\sqrt{-1}\), you can manage calculations involving square roots of negatives efficiently. When you encounter a problem involving the square root of a negative, just think of it in terms of \(i\), and reframe the problem so that it can be simplified like you would with typical positive numbers.

The Process of Factoring

Factoring is a method used to break down numbers into their multiples. It's particularly helpful when simplifying expressions under a square root. For example, consider the number 28. It can be factored into 4 and 7 (since \(28 = 4 \times 7\)), where 4 is a perfect square. By identifying perfect squares within the factors, we can simplify the square root of the number further.
In the exercise about \(\sqrt{-28}\), after rewriting it as \(\sqrt{-1} \times \sqrt{28}\), the factoring of 28 into 4 and 7 enables you to simplify \(\sqrt{28}\) to \(2\sqrt{7}\). Once factorization is complete, combining the terms with \(i\) results in the expression \(2i\sqrt{7}\). Factoring, therefore, makes it easier to work with larger numbers and simplifies complex expressions significantly.