Problem 28 - Algebra for College Students

Question

Find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers. \(\sqrt{2 x}(\sqrt{12 x y}-\sqrt{8 y})\)

Step-by-Step Solution

Answer

The product is \\( 2x\sqrt{6y} - 4\sqrt{xy} \\).

Step 1: Distribute the square root

Start by distributing the \( \sqrt{2x} \) to each term inside the parentheses: \( \sqrt{2x}(\sqrt{12x y}) \) and \( \sqrt{2x}(-\sqrt{8y}) \).

Step 2: Multiply the square roots

Use the property that \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \) to simplify each product. 1. \( \sqrt{2x} \cdot \sqrt{12x y} = \sqrt{(2x)(12x y)} = \sqrt{24x^2 y} \).2. \( \sqrt{2x} \cdot (-\sqrt{8y}) = -\sqrt{(2x)(8y)} = -\sqrt{16xy} \).

Step 3: Simplify each square root

Simplify by finding perfect squares within each square root:1. \( \sqrt{24x^2 y} = \sqrt{(4)(6)(x^2)(y)} = 2x\sqrt{6y} \).2. \( -\sqrt{16xy} = -4\sqrt{xy} \), since \( 16 \) is \( 4^2 \).

Step 4: Combine the simplified expressions

Combine the simplified expressions to get the final result: \( 2x\sqrt{6y} - 4\sqrt{xy} \).

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic property used to simplify expressions. It comes into play when you have an expression like \( a(b + c) \), which can be expanded to \( ab + ac \). This property helps in distributing one term across terms inside a parenthesis, allowing for simplification or expansion of the expression.

In the original exercise, \( \sqrt{2x} \) is outside the parenthesis, requiring its distribution across the terms inside. So, \( \sqrt{2x}(\sqrt{12xy} - \sqrt{8y}) \) becomes \( \sqrt{2x} \times \sqrt{12xy} \) and \( \sqrt{2x} \times (-\sqrt{8y}) \).

By distributing the term effectively, you can manipulate the equation into a form that is simpler and easier to work with.

Multiplying Radicals

When multiplying radicals, an important principle is that \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \). This means you can combine the numbers inside the square roots before evaluating the radical. This is particularly useful for simplifying more complex expressions.

In our problem, we multiply \( \sqrt{2x} \times \sqrt{12xy} \) becoming \( \sqrt{24x^2y} \), and \( \sqrt{2x} \times (-\sqrt{8y}) \) turning into \( -\sqrt{16xy} \).

Carefully apply this property to handle radicals confidently, ensuring that variables and coefficients inside the square roots are carefully considered.

Simplifying Square Roots

Simplifying square roots involves breaking down the expression under the square root into its simplest form. Identifying perfect squares within the factorization aids in this process. Perfect squares like \( 4, 9, 16 \), etc., simplify easily because their square roots are whole numbers.

For example, \( \sqrt{24x^2y} \) can be rewritten as \( \sqrt{4 \times 6 \times x^2 \times y} \). This allows us to take out \( 2x \) since \( \sqrt{4} = 2 \) and \( \sqrt{x^2} = x \), leaving \( 2x\sqrt{6y} \).

Similarly, \( -\sqrt{16xy} \) simplifies directly to \( -4\sqrt{xy} \), because \( \sqrt{16} = 4 \).

Breaking down the components this way is crucial in presenting the solution in its simplest radical form.

Perfect Squares

Perfect squares are numbers whose square roots are integers. Recognizing these within expressions is vital as they simplify calculations drastically. Numbers like \( 1, 4, 9, 16, 25 \) are common examples.When simplifying square roots, spotting perfect squares helps pluck out constants. For example, from \( \sqrt{16xy} \), recognize that \( 16 \) is a perfect square, and thus \( \sqrt{16} = 4 \). Thus, \( -\sqrt{16xy} \) becomes \( -4\sqrt{xy} \).Another instance is in \( \sqrt{24x^2y} \), where the expression contains \( x^2 \), a perfect square, meaning \( \sqrt{x^2} = x \). This process simplifies the expression efficiently, making it manageable.