Problem 20 - Algebra 1

Question

Simplify the expression. $$\sqrt{2} \cdot \sqrt{8}$$

Step-by-Step Solution

Answer

The simplified result of the expression is 4

Step 1: Express the given expression

The given expression is \( \sqrt{2} \cdot \sqrt{8} \).

Step 2: Simplify square root of 8

The square root of 8 can be reduced by breaking 8 down into \(2^3\). So, \( \sqrt{8} = \sqrt{2^3} = 2\sqrt{2}\), because the square root of \(2^2\) is 2.

Step 3: Apply the simplified square root

Substitute \(2\sqrt{2}\) for \(\sqrt{8}\) to get \( \sqrt{2} \cdot 2\sqrt{2} \).

Step 4: Simplify the expression

The square root of 2 times the square root of 2 is 2. As such, the expression can be simplified further into \(2 \cdot 2 = 4\).

Key Concepts

Square Roots

Square roots are essential to simplify expressions, especially when dealing with radical expressions. A square root represents a value that, when multiplied by itself, gives the original number. For instance, the square root of 4 is 2 because when 2 is multiplied by itself, it results in 4.

Square roots are represented by the symbol \(\sqrt{}\). When simplifying expressions like \(\sqrt{8}\), breaking the number into its prime factors helps. In this example, 8 can be expressed as \(2^3\), allowing it to be simplified as \(2\sqrt{2}\).

Understanding the fundamentals of square roots helps in performing proper simplifications. In the expression \(\sqrt{2} \times \sqrt{8}\), breaking down square roots into simpler parts can lead to the easier combination of terms.

Properties of Radicals

Radicals, often depicted by the square root symbol, come with a set of properties that simplify mathematical expressions. These properties make it possible to manipulate and reduce expressions effectively.

• Product Property: This property states that \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\). For example, \(\sqrt{2} \cdot \sqrt{8}\) can be written as \(\sqrt{16}\), which simplifies to 4.
• Power Property: If a number is expressed in squared terms, \(\sqrt{a^2} = a\). When simplifying \(\sqrt{8}\), which is \(\sqrt{2^3}\), the \(2^2\) component can be extracted as 2, resulting in \(2\sqrt{2}\).

Knowing these properties will allow you to efficiently break down and combine radical expressions, simplifying problems efficiently.

Mathematical Operations with Radicals

Performing operations with radicals involves some simple rules and manipulations. Combining these can offer a complete picture of how to handle such expressions.

When multiplying radicals, like in the example \(\sqrt{2} \cdot \sqrt{8}\), remember that the properties of radicals can simplify your work. First, transform each radical into its simplest form and apply the product property. Here, replacing \(\sqrt{8}\) with \(2\sqrt{2}\) turns the operation into \(\sqrt{2} \times 2\sqrt{2}\).

After applying the product property for the simplification \(\sqrt{2} \times \sqrt{2} = 2\), the expression becomes \(2 \cdot 2\). Thus, this expression simplifies to 4. By breaking down each step, understanding operations with radicals becomes both clear and manageable.