Question

Simplify each expression. \(\sqrt{2} \cdot \sqrt{6}\)

Short answer

The simplified expression is \(2\sqrt{3}\).

Step-by-step solution

Understand the Property

Recognize that the expression involves the product of two square roots: \(\sqrt{2}\) and \(\sqrt{6}\). We can use the property of square roots which is \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\).

Apply the Property

Apply the property from Step 1 to simplify the expression: \(\sqrt{2} \cdot \sqrt{6} = \sqrt{2 \times 6}\).

Perform the Multiplication Inside the Square Root

Multiply the numbers inside the square root: \(\sqrt{2 \times 6} = \sqrt{12}\).

Simplify the Square Root

Simplify \(\sqrt{12}\). Notice that 12 can be factored into 4 and 3: \(12 = 4 \times 3\). Therefore, \(\sqrt{12} = \sqrt{4 \times 3}\).

Separate into Two Square Roots

Separate \(\sqrt{12}\) into two square roots: \(\sqrt{4} \times \sqrt{3}\).

Simplify Individual Square Roots

Recognize that \(\sqrt{4} = 2\), so the expression becomes \(2 \times \sqrt{3}\).

Finalize the Simplified Expression

The simplified form of the original expression \(\sqrt{2} \cdot \sqrt{6}\) is \(2\sqrt{3}\).

Key concepts

Properties of Square Roots

Understanding the properties of square roots is fundamental to simplifying expressions involving square roots. One of the most useful properties is that the square root of a product is equal to the product of the square roots of each factor.

This property can be mathematically represented as: \[ \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \] This property allows us to break down complex square roots into simpler components, making them easier to handle.

For example, if you have the expression \( \sqrt{2} \cdot \sqrt{6} \), by using this property, you can combine it into a single square root: \[ \sqrt{2 \cdot 6} = \sqrt{12} \]This simplifies the process of evaluating or further simplifying the expression.

Multiplication of Square Roots

Multiplying square roots can be straightforward when you apply the right property. When you encounter something like \( \sqrt{2} \cdot \sqrt{6} \), the first step is to recognize the multiplication property of square roots.

Observe that: \[ \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \] Here, we multiply the numbers under the square roots directly before continuing to simplify the expression. With \( \sqrt{2} \cdot \sqrt{6} \), you simply compute \( 2 \times 6 = 12 \) and rewrite the expression as \( \sqrt{12} \).

This process combines the square roots into a single expression, which can be further simplified if possible, as seen in the next steps.

Simplifying Radical Expressions

Simplifying radical expressions is a vital skill that enables you to express square roots in their simplest form. After multiplying the numbers under the square root, the next step is to break it down into simpler factors.

For example, if you end up with \( \sqrt{12} \), seek the largest square factor: \[ 12 = 4 \times 3 \] Next, use the property of square roots to split \( \sqrt{12} \) into its components: \[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \cdot \sqrt{3} \] Then, simplify \( \sqrt{4} \) to 2, leading to \( 2 \cdot \sqrt{3} \).

This results in the simplest form \( 2\sqrt{3} \), giving you a clearer and more concise way to express the original problem. Remember always to look for perfect squares in your product to simplify your radical expressions efficiently.