Question

Simplify each expression. \(\sqrt{8} \cdot \sqrt{9}\)

Short answer

The simplified expression is \(6\sqrt{2}\).

Step-by-step solution

Apply the Product Property of Square Roots

The Product Property of Square Roots states that the square root of a product is equal to the product of the square roots: \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\). We can combine the terms \(\sqrt{8} \cdot \sqrt{9}\) under one square root: \(\sqrt{8 \cdot 9}\).

Multiply the Numbers Under the Square Root

Multiply the numbers inside the square root: \(8 \cdot 9 = 72\). Hence, the expression becomes \(\sqrt{72}\).

Simplify the Square Root

Identify a perfect square factor of 72. The number 72 can be factored into \(36 \times 2\). Since 36 is a perfect square, we can simplify \(\sqrt{72}\) to \(6\sqrt{2}\) because \(\sqrt{36} = 6\).

Key concepts

Product Property of Square Roots

When dealing with square roots, the product property is a valuable tool to keep things simple. This property states: \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\). It allows us to divide a single square root expression into two separate roots or combine two square roots into one. This is particularly useful for simplification.

• For instance, \(\sqrt{8} \cdot \sqrt{9}\) can be combined using this property: \(\sqrt{8 \cdot 9}\).
• It's essential for understanding multiplication within radical expressions and paves the way for further simplification.

Remember, to use the product property effectively, the numbers involved must be non-negative to ensure the result remains valid for real numbers.

Perfect Square Factorization

Perfect squares play a crucial role in simplifying square roots. A perfect square is a number that can be expressed as the square of an integer, such as 4, 9, 16, and so on.

• When simplifying a square root expression like \(\sqrt{72}\), look for perfect square factors. In this case, 72 can be factored into \(36 \times 2\).
• Since 36 is a perfect square, \(\sqrt{36} = 6\). Thus, the expression becomes \(6\sqrt{2}\).

Identifying such factors allows us to pull these perfect squares out of the radical, simplifying the expression to its simplest form. It's an essential skill for handling complex radical expressions.

Arithmetic Operations in Radical Expressions

Arithmetic operations like addition, subtraction, multiplication, and division are possible with radical expressions and follow specific rules.

• When multiplying, use the product property of square roots to simplify first. For example, \(\sqrt{8} \cdot \sqrt{9}\) becomes \(\sqrt{72}\).
• Simplify further by breaking down \(\sqrt{72}\) using perfect square factorization, achieving the simplest form \(6\sqrt{2}\).

While addition and subtraction require like terms (same radicand), multiplication and division rely heavily on properties of roots to ensure correct simplification. Always aim for the simplest form to achieve clarity and ease in calculations.