Problem 39 - Algebra for College Students

Question

Write each of the following in terms of \(i\) and simplify. For example, $$ \sqrt{-20}=i \sqrt{20}=i \sqrt{4} \sqrt{5}=2 i \sqrt{5} $$ $$-2 \sqrt{-80}$$

Step-by-Step Solution

Answer

The simplified expression is \(-8i\sqrt{5}\).

Step 1: Identify the Negative Under the Square Root

The expression given is \(-2 \sqrt{-80}\), which indicates we are dealing with the square root of a negative number. Recognizing this allows us to know that we must utilize the imaginary unit \(i\), where \(i = \sqrt{-1}\).

Step 2: Rewrite the Expression with \(i\)

Since the square root includes a negative number, rewrite \(\sqrt{-80}\) as \(i \sqrt{80}\). The expression now becomes \(-2 \times i \sqrt{80}\).

Step 3: Simplify the Square Root Term

Next, we need to break down \(\sqrt{80}\) into simpler square root factors. We know that \(80 = 16 \times 5\), so \(\sqrt{80} = \sqrt{16 \times 5} = \sqrt{16} \cdot \sqrt{5} = 4\sqrt{5}\).

Step 4: Substitute Back and Simplify

Substitute \(4\sqrt{5}\) back into the original expression to get \(-2 \times i \times 4\sqrt{5} = -2 \times 4 \times i \sqrt{5} = -8i\sqrt{5}\). The expression has now been simplified.

Key Concepts

Square Roots

When we talk about square roots, we're essentially looking for a number that, when multiplied by itself, gives us a specific original number. For example, the square root of 9 is 3, because 3 times 3 equals 9. Square roots are not just about positive numbers. Sometimes we encounter the square root of negative numbers. This is where imaginary numbers come into play, introducing the imaginary unit denoted as \(i\), where \(i = \sqrt{-1}\). This helps us work with square roots of negative numbers, which would otherwise be undefined in the realm of real numbers. In our exercise,
\(\sqrt{-80}\) is simplified using this concept: we first write it as \(i\sqrt{80}\) because of the negative sign under the square root. This step is crucial in transitioning from real to complex number calculations.

Simplifying Expressions

Simplifying expressions is about breaking down mathematical expressions into their simplest form. It makes calculations easier and helps in visualizing what the problem represents. For the expression in the exercise, \(-2 \sqrt{-80}\) becomes \(-2 \times i \sqrt{80}\).
The next step in simplification is breaking down \(\sqrt{80}\) into its prime factors. We do this by finding perfect square factors. Here, \(80\) can be expressed as \(16 \times 5\), where 16 is a perfect square. Therefore, \(\sqrt{80}\) simplifies to \(\sqrt{16} \cdot \sqrt{5}\), leading us to {{"4\sqrt{5}".
By identifying perfect squares within a number, we can simplify square roots, making calculations with them more approachable.

Algebraic Expressions

Algebraic expressions use numbers, variables, and operations to represent a mathematical situation. In this context, working with expressions like \(-2 \times i \times 4\sqrt{5}\) involves applying algebraic principles to simplify the components systematically. Each part of the expression has a role: the \(-2\) is a constant coefficient, \(i\) is the imaginary unit representing the square root of \(-1\), and \(4\sqrt{5}\) is the simplified form of the square root component.Let's break down the expression:

• -2: The expression starts with this negative coefficient.
• i: Introduced to manage the negative square root.
• 4\sqrt{5}: The simplified square root of 80.

Together, these components form \(-8i\sqrt{5}\), which is the simplified version. Simplifying algebraic expressions often requires rewinding complex operations into recognizable forms, making them easier to interpret and solve.