Problem 9 - College Algebra with Modeling and Visualization

Question

Simplify by using the imaginary unit \(i\). $$ \sqrt{-54} $$

Step-by-Step Solution

Answer

The simplified form is \(3i \sqrt{6}\).

Step 1: Express with imaginary unit

To express the square root of a negative number, we use the imaginary unit. Recall that \(i\) is defined as \(i = \sqrt{-1}\) which implies \(i^2 = -1\). Hence, we can write \(\sqrt{-54} = \sqrt{54} \cdot \sqrt{-1} = \sqrt{54} \cdot i\).

Step 2: Simplify the square root of a positive number

Next, we need to simplify \(\sqrt{54}\). Breaking it down, we note that 54 can be factored into 9 and 6, where 9 is a perfect square. Therefore, \(\sqrt{54} = \sqrt{9 \times 6} = \sqrt{9} \times \sqrt{6} = 3 \sqrt{6}\).

Step 3: Combine the simplified terms

Now, combine what we have from Steps 1 and 2. The expression becomes \(3 \sqrt{6} \cdot i\). Hence, the simplification of \(\sqrt{-54}\) is \(3i \sqrt{6}\).

Key Concepts

Imaginary unit

When you encounter the square root of a negative number, things can become a bit tricky because real numbers don't have square roots that are negative. This is where the concept of the imaginary unit comes into play. The imaginary unit is represented by the symbol \( i \), where \( i = \sqrt{-1} \).
Understanding \( i \) helps us handle negative square roots:

• Negative numbers under a square root sign can be rewritten as a product of \( i \) and the square root of the positive counterpart.
• For example, \( \sqrt{-54} \) can be split into \( \sqrt{54} \times \sqrt{-1} \), simplifying to \( \sqrt{54} \times i \).

This method makes it possible to simplify expressions involving negative square roots, transforming a potentially complex problem into a form that's much easier to manage.

Square root simplification

To simplify expressions like \( \sqrt{54} \), it's important to break the number down into its component factors. Let's explore how we can do this:
54 can be factored into 9 and 6, because \( 54 = 9 \times 6 \). Recognizing that 9 is a perfect square is crucial:

• Since \( 9 = 3^2 \), we know that \( \sqrt{9} = 3 \).
• Thus, \( \sqrt{54} = \sqrt{9 \times 6} = \sqrt{9} \times \sqrt{6} = 3 \sqrt{6} \).

Finding perfect squares within the factorization of a number makes simplification straightforward. This technique is particularly helpful in expressions involving imaginary units, as it reduces complexity and aids in combining terms effortlessly.

Perfect squares

Perfect squares are numbers that can be expressed as the square of an integer. For instance, numbers like 1, 4, 9, 16, and so on are all perfect squares. Let's break down why perfect squares are helpful in simplifying square roots:
Cleverly identifying them in factorization problems can dramatically simplify expressions.

• If you recognize a perfect square within a factor, you can simplify the square root easily. For instance, since 9 is a perfect square (\(9 = 3^2\)), we know \(\sqrt{9} = 3\).
• This transforms \( \sqrt{54} \) into \( 3 \sqrt{6} \), as we simplify by pulling the square root of the perfect square factor out.

Understanding perfect squares is a significant skill for dealing with both real and complex numbers. This knowledge is useful when simplifying expressions fluently and understanding deeper mathematical concepts.