Problem 18 - College Algebra with Modeling and Visualization

Question

Simplify by using the imaginary unit \(i\). $$ \sqrt{-20} \cdot \sqrt{-5} $$

Step-by-Step Solution

Answer

The expression simplifies to \(-10\).

Step 1: Understand the Imaginary Unit

The imaginary unit, represented by \(i\), is defined as \(i = \sqrt{-1}\). This definition is important because it allows us to express the square root of any negative number \(a\) as \(\sqrt{-a} = i\sqrt{a}\).

Step 2: Express Each Square Root with \(i\)

Rewrite each square root using the imaginary unit. \(\sqrt{-20}\) becomes \(i\sqrt{20}\) and \(\sqrt{-5}\) becomes \(i\sqrt{5}\). This is because both expressions involve the square root of a negative number.

Step 3: Simplify the Product

Compute the product of \(\sqrt{-20} \cdot \sqrt{-5}\) by substituting the expressions from Step 2: \[(i\sqrt{20}) \cdot (i\sqrt{5}) = i^2 \cdot \sqrt{20} \cdot \sqrt{5}\]

Step 4: Simplify Using \(i^2 = -1\)

Recall that \(i^2 = -1\). Substitute -1 for \(i^2\) in the expression: \[\sqrt{-20} \cdot \sqrt{-5} = -\sqrt{20}\cdot \sqrt{5}\]

Step 5: Simplify the Radicals

Calculate \(\sqrt{20} \cdot \sqrt{5} = \sqrt{100}\) because multiplying two square roots \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\). Then, \(\sqrt{100} = 10\).

Step 6: Combine and Simplify

Substitute the simplified product back into the expression: \(-\sqrt{100} = -10\). Thus, the original expression simplifies to \(-10\).

Key Concepts

Imaginary Unit

The imaginary unit is a fundamental concept in complex numbers. This unit is represented by the symbol \(i\). The key property of \(i\) is that it is defined by \(i = \sqrt{-1}\). This definition allows mathematicians to deal with the square roots of negative numbers, which otherwise would be undefined in the realm of real numbers.

When you use \(i\), you can express negative square roots as simpler expressions involving \(i\). For example, if you have \(\sqrt{-a}\), you can rewrite it as \(i\sqrt{a}\). This is a very useful tool because it simplifies operations involving negative square roots and gives us the means to expand into the broader field of complex numbers.When working with the imaginary unit, always keep in mind that the square of \(i\) is \(-1\), i.e., \(i^2 = -1\). This property comes in handy for operations like multiplication, as it essentially allows you to convert a part of your expression into a real number, facilitating further simplification.

Simplifying Expressions

Simplifying expressions using the imaginary unit requires a clear understanding of mathematical properties. When you encounter complex expressions, especially those involving square roots of negative numbers, your goal should be to write them as simply as possible.

Start by using the imaginary unit to express square roots of negative numbers. Next, recognize any identities or properties that can help simplify further. For instance, in our exercise:

• You express \(\sqrt{-20}\) as \(i\sqrt{20}\) and \(\sqrt{-5}\) as \(i\sqrt{5}\).
• Multiply these to find \(\sqrt{-20} \cdot \sqrt{-5} = (i\sqrt{20}) \cdot (i\sqrt{5})\).
• Knowing that \(i^2 = -1\), this becomes \(-\sqrt{20} \cdot \sqrt{5}\).

In simplifying further, you multiply the numerical roots together: \(\sqrt{20} \cdot \sqrt{5} = \sqrt{100} = 10\), hence the expression becomes \(-10\). Remember to apply the properties of radicals, such as \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\), to streamline calculations.Always make sure your final answer is as simple as possible. If an expression involves complex numbers, try to ensure it's in a standard form, typically with any \(i\)-related terms handled first.

Square Roots of Negative Numbers

Dealing with square roots of negative numbers is a pivotal aspect of complex numbers. In real-number calculations, the square root of a negative number is not defined. However, the introduction of the imaginary unit \(i\) allows us to reframe these calculations.

For any negative number \(-a\), its square root is expressed in terms of \(i\) as \(\sqrt{-a} = i\sqrt{a}\). This conversion simplifies many mathematical operations and gives us new tools to consider problems that involve these roots.Let's say you have an expression \(\sqrt{-b}\) as part of a calculation. By using \(i\), you can rewrite this as \(i\sqrt{b}\). This enables you to perform multiplications, divisions, and other algebraic operations more straightforwardly. Consider the key steps:

• Identify the negative square root in an expression.
• Rewrite it using the imaginary unit as \(i\sqrt{positive\; part}\).
• Simplify the expression as necessary, often using \(i^2 = -1\).

Moreover, learning to manage these forms gives you a strong foundation for delving deeper into complex number systems, where you frequently encounter the square roots of negative numbers. It opens up avenues for exploring equations and systems that are more expansive than those limited to real numbers alone.