Problem 3 - 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. $$ \sqrt{2} \cdot \sqrt{8} $$

Step-by-Step Solution

Answer

The product simplifies to 4.

Step 1: Apply Product Property of Square Roots

We know that the product of square roots can be combined under a single square root. Apply the property: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). Therefore, \( \sqrt{2} \cdot \sqrt{8} = \sqrt{2 \times 8} \).

Step 2: Calculate the Product under the Square Root

Multiplying the values inside the square root gives us: \( 2 \times 8 = 16 \). Therefore, the expression simplifies to \( \sqrt{16} \).

Step 3: Simplify the Expression

Find the square root of 16. Since 16 is a perfect square, \( \sqrt{16} = 4 \).

Step 4: Finalize the Simplified Expression

The expression \( \sqrt{2} \cdot \sqrt{8} \) simplifies to 4. Therefore, the product in its simplest form is 4.

Key Concepts

Simplifying Square Roots

Simplifying square roots involves expressing a square root in its most reduced form, which makes calculations much easier. When you simplify square roots, you're essentially looking for common factors of the number inside the square root (known as the radicand) that are perfect squares.
For example, in the exercise given, we have to simplify the product of two square roots: \(\sqrt{2} \cdot \sqrt{8}\). By using the product property of square roots, we can combine these into one square root. The challenge then becomes finding the simplest form of this new square root, which is done by checking if the radicand has perfect square factors. This allows us to break it down and simplify effectively.
Understanding this process is crucial as it plays a foundational role in higher mathematical calculations and problem-solving.

Perfect Squares

Perfect squares are numbers that result from squaring an integer. Recognizing perfect squares is vital for simplifying square roots, because it allows you to immediately know the square root of the number without further calculation.
In our example of \(\sqrt{16}\), 16 is a perfect square because it equals \(4^2\). This immediately tells us that \(\sqrt{16} = 4\).
Common perfect squares that are useful to memorize include:

• \(1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144\) and so on.

When simplifying square roots, breaking down the radicand to its largest perfect square factor is helpful, as it simplifies the entire process and helps in finding the reduced form of a square root quickly.

Product Property of Square Roots

The product property of square roots is a fundamental principle that allows the multiplication of square roots to be simplified by combining them into a single square root. The property states: \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\).
Applying this property in the given exercise, we went from \(\sqrt{2} \cdot \sqrt{8}\) to \(\sqrt{16}\). By multiplying the numbers under the square root sign, you streamline the simplification process.
This property is invaluable not just for simplifying numbers, but also when dealing with algebraic expressions. It simplifies calculations and is especially helpful when dealing with large numbers or variables within square roots.

Radicand

The radicand is the number inside the square root symbol that you are finding the square root of. Recognizing the radicand is crucial because simplifying it often holds the key to solving square root problems effectively.
In the given exercise \(\sqrt{2} \cdot \sqrt{8}\), after applying the product property, the radicand becomes 16: \(\sqrt{16}\) is what we simplify.
To break down and simplify a square root, look at the radicand's factors, especially perfect squares. Understanding how to manipulate the radicand can help you simplify complex square roots and make numbers much more manageable.