Question

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

Short answer

\( \sqrt{2} \cdot \sqrt{2} = 2 \)

Step-by-step solution

Understand the Operation

The expression involves two square roots multiplied together: \( \sqrt{2} \cdot \sqrt{2} \). Multiplying two square roots is analogous to taking the square root of the product of the numbers inside.

Apply the Product Rule for Square Roots

The product rule for square roots states that \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). Therefore, we can rewrite \( \sqrt{2} \cdot \sqrt{2} \) as \( \sqrt{2 \cdot 2} \).

Simplify the Product Inside the Square Root

Multiply the numbers inside the square root: \( 2 \times 2 = 4 \). Hence, the expression becomes \( \sqrt{4} \).

Calculate the Square Root

The square root of 4 is a simple calculation: \( \sqrt{4} = 2 \). Therefore, the expression simplifies to 2.

Key concepts

Product Rule for Square Roots

The product rule for square roots is a useful mathematical property that allows us to simplify expressions involving the multiplication of square roots. According to this rule, when you multiply two square roots together, you can combine them under a single square root. The rule is mathematically expressed as: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). This rule helps in simplifying expressions by reducing the number of square root operations needed.

• If you have \( \sqrt{2} \cdot \sqrt{3} \), you can rewrite it as \( \sqrt{2 \cdot 3} = \sqrt{6} \).
• The product rule only applies if multiplying square roots with positive numbers.

Using this rule in your calculations can simplify processes in algebra, making it easier to deal with more complex expressions.

Square Roots

A square root is a value that, when multiplied by itself, gives the original number. The square root symbol \( \sqrt{} \) is used to denote this operation. For example:

• The square root of 4 is 2 because \( 2 \times 2 = 4 \).
• Similarly, the square root of 9 is 3 since \( 3 \times 3 = 9 \).

Understanding square roots is crucial in many areas of mathematics, especially when dealing with quadratic equations and other algebraic expressions. It's important to remember that every positive number has two square roots — one positive and one negative. However, in most real-world scenarios, we use the positive square root.
Simplifying square roots often involves using rules or properties such as the product rule to make calculations easier.

Multiplication of Square Roots

Multiplying square roots is a straightforward process once you understand the product rule for square roots. It's essentially about grouping the numbers beneath a single square root and then simplifying:

• For example, if you multiply \( \sqrt{2} \) by \( \sqrt{3} \), you use the product rule to combine them into \( \sqrt{6} \).
• If you multiply two identical square roots, like \( \sqrt{2} \cdot \sqrt{2} \), you get \( \sqrt{4} \), which simplifies to 2.

Multiplying square roots is a frequent task in algebra, and mastering this skill can greatly enhance your problem-solving abilities. Just remember to always simplify the final expression to its simplest form. This step ensures that your answer is as clean and understandable as possible.