Problem 29 - 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. $$ \sqrt{12 x y^{3}}(\sqrt{3 x y}+3) $$

Step-by-Step Solution

Answer

The simplest form is \( 6xy^2 + 6y \sqrt{3xy} \).

Step 1: Simplify each square root term

Simplify each square root term by factoring under the square root to find perfect squares. For \( \sqrt{12xy^3} \): - Factor: \( 12 = 4 \times 3 \), \( y^3 = y^2 \times y \), giving \( \sqrt{4 \cdot 3xy^2 \cdot y} \).- Simplified: \( \sqrt{4} = 2 \) and \( \sqrt{y^2} = y \). This results in \( 2y\sqrt{3xy} \).For \( \sqrt{3xy} \): - Cannot be simplified further as there are no perfect squares.

Step 2: Distribute to each term inside the parentheses

Distribute \( \sqrt{12xy^3} = 2y\sqrt{3xy} \) to each term inside the parentheses: - First, multiply \( 2y \sqrt{3xy} \times \sqrt{3xy} \): \[ = 2y \sqrt{(3xy)(3xy)} = 2y \sqrt{9x^2y^2} = 2y \times 3xy = 6xy^2 \]- Next, multiply \( 2y \sqrt{3xy} \times 3 \): \[ = 6y \sqrt{3xy} \]

Step 3: Combine and write in simplest form

Combine the results from the distribution step:The expression becomes:\[ 6xy^2 + 6y \sqrt{3xy} \] This expression cannot be simplified further, as the terms cannot be combined.

Key Concepts

Perfect Squares

A perfect square is the product of a number multiplied by itself. In mathematics, perfect squares simplify calculations, especially within radical expressions. They are key in reducing square roots to their simplest form. For example, 4 is a perfect square because it equals \(2 \times 2\). Similarly, \(y^2\) is a perfect square, resulting from \(y \times y\). Recognizing these helps simplify expressions, such as \(\sqrt{12}\), broken down into \(\sqrt{4 \times 3}\), simplifying to \(2\sqrt{3}\). Adopting this approach makes it easier to work with radicals.

Distributive Property

The distributive property involves multiplying a term outside a set of parentheses by each term within the parentheses. This principle is crucial in algebra for expanding expressions. In the exercise, we apply it to \(\sqrt{12xy^3}(\sqrt{3xy} + 3)\). You distribute \(2y\sqrt{3xy}\), obtained from the earlier simplification, to both terms inside the parentheses. This expands the expression, helping break down complex problems into manageable parts. Keeping this in mind allows a more straightforward solution strategy, enhancing understanding.

Square Roots

Square roots are inverses of squaring a number, representing one of two equal factors. For instance, the square root of 9, represented as \(\sqrt{9}\), is 3, because \(3 \times 3 = 9\). Simplifying square roots, especially in radical expressions, frequently involves identifying perfect squares within the radicand (the number inside the root symbol). In algebra, understanding these principles becomes crucial when manipulating square roots to simplify expressions by isolating identifiable factors, such as in the simplification \(\sqrt{12xy^3}\) to \(2y\sqrt{3xy}\). This ensures clarity and accuracy in solutions.

Algebra 2

Algebra 2 builds upon foundational algebra by introducing more complex concepts, including radical expressions. Understanding simplification requires recognizing patterns, such as factoring into perfect squares, and using properties like the distributive property. These skills are applied in exercises like the original, where combined methods are necessary to simplify complex algebraic expressions. Through these exercises, students develop problem-solving strategies and enhance their algebraic agility, which is essential for tackling advanced mathematical problems.