Problem 14 - Algebra 2 and Trigonometry

Question

In \(3-41\) , express each product in simplest form. Variables in the radicand with an even index are non-negative. $$ (-2 \sqrt{5})^{2} $$

Step-by-Step Solution

Answer

The simplest form of \((-2 \sqrt{5})^{2}\) is 20.

Step 1: Understand the Expression

We have the expression \((-2 \sqrt{5})^{2}\). This implies we will square everything inside the parentheses: \(-2\) and \(\sqrt{5}\).

Step 2: Square the Coefficient

Square the coefficient \(-2\). Calculating \((-2)^{2}\), we get: \[ (-2)^{2} = 4 \]

Step 3: Square the Radical Part

Now, square the radical part \(\sqrt{5}\). Squaring \(\sqrt{5}\), we get: \[ (\sqrt{5})^{2} = 5 \]

Step 4: Multiply the Results

Multiply the results from Steps 2 and 3. We have: \[ 4 \times 5 = 20 \]

Step 5: Express in Simplest Form

The expression \((-2 \sqrt{5})^{2}\) simplifies to 20.

Key Concepts

Squaring Radicals

Understanding how to square a radical expression is essential in algebra. A radical expression involves a root, such as the square root. When squaring the radical part of an expression, like \(\sqrt{5}\), you essentially reverse the square root operation. By definition, when you square a square root, you "cancel out" the square root symbol, leaving the value under the root. Here's what happens in our example:

• Given \((\sqrt{5})^{2}\), the square and the square root neutralize each other.
• The result is simply the radicand itself: \(5\).

This is because the square of a square root always returns you to the original number under the radical.

Squaring Integers

Squaring an integer involves multiplying the number by itself. This is straightforward: take the number and find the product of multiplying it by itself. Let's look closely at how we apply this to our example:

• Consider the integer coefficient \(-2\) in the expression \((-2 \sqrt{5})^{2}\).
• Square it by calculating \((-2)\times (-2)\), which results in \(4\).

The negative sign also plays an important role here. Recall that the square of any negative number becomes positive, because multiplying two negative numbers results in a positive product.

Multiplying Results

After squaring each part of the expression separately, the next step is to combine those results. This involves multiplying the squared integer and the squared radical part together. Here's how that works in this context:

• We've determined from our previous steps that \((-2)^{2} = 4\) and \((\sqrt{5})^{2} = 5\).
• Multiply these results: \(4 \times 5\).
• This gives us the product of \(20\).

The multiplication of these two straightforward components brings you to the simple result of the entire expression.

Expressing in Simplest Form

The final step in simplifying the expression involves presenting your result in its simplest form. Simplification means reducing your answer to its most basic version, without any radicals or unnecessary components. In our case, the initial expression was \((-2 \sqrt{5})^{2}\). After processing each part of the expression separately:

• We squared all components, yielding an integer result.
• The final product obtained was \(20\).

Simplifying an expression allows it to be more easily understood and used in further calculations. Once you reach a final, non-radical result like this, you know the expression is in its simplest form.