Problem 59 - Intermediate Algebra

Question

Multiply and then simplify if possible. $$ \sqrt{2}(\sqrt{2}+x \sqrt{6}) $$

Step-by-Step Solution

Answer

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

Step 1: Distribute the Square Root

Apply the distributive property by multiplying \( \sqrt{2} \) with each term inside the parentheses: \( \sqrt{2} \times (\sqrt{2} + x \sqrt{6}) \).

Step 2: Multiply the Terms

Multiply each term individually, starting with \( \sqrt{2} \times \sqrt{2} \), which equals \( 2 \), and then \( \sqrt{2} \times x \sqrt{6} \), which equals \( x\sqrt{12} \).

Step 3: Simplify the Expression

The expression now is \( 2 + x\sqrt{12} \). Simplify \( \sqrt{12} \) to \( \sqrt{4 \times 3} = 2\sqrt{3} \). Hence, the expression becomes \( 2 + 2x\sqrt{3} \).

Step 4: Write the Final Simplified Expression

Combine all terms to get the final expression \( 2 + 2x\sqrt{3} \).

Key Concepts

Distributive Property

Distributive property is a fundamental concept in mathematics that helps us multiply a single term across a sum or difference within parentheses. Imagine it as sharing the term outside the parentheses with each term inside. In our exercise, this means taking \( \sqrt{2} \) and multiplying it by each individual term within the brackets: \( (\sqrt{2} + x \sqrt{6}) \).

By applying the distributive property here, we'll do two separate multiplications:

• First, \( \sqrt{2} \times \sqrt{2} \)
• Then, \( \sqrt{2} \times x \sqrt{6} \)

Once distributed, each term can be simplified separately, setting the foundation for dealing with more complex expressions. This step makes it easier to handle the elements break down and recombine them clearly.

Simplification of Radicals

Simplifying radicals involves rewriting them in a form that's easier to understand or use. Radicals are expressions that contain a square root, cube root, etc. In our problem, we encounter \( \sqrt{12} \).

To simplify a radical like \( \sqrt{12} \), we look for perfect squares that divide evenly into the number under the root.

• Since \( \sqrt{12} = \sqrt{4 \times 3} \), we know that \( \sqrt{4} = 2 \).
• Therefore, \( \sqrt{12} \) simplifies to \( 2\sqrt{3} \).

This simplification allows us to express the radical in a more refined way, making any calculations much less cumbersome. Once simplified, it's easier to see relationships between numbers and perform further algebraic operations accurately.

Square Roots

Square roots are a way to find a number which, when multiplied by itself, gives the original number. In mathematical terms, if \( a^2 = b \), then \( a \) is called the square root of \( b \). In the context of our problem, square roots help us manage complex expressions with ease.

Understanding square roots simplifies the process of multiplying them. For example, when multiplying two square roots like \( \sqrt{2} \times \sqrt{2} \), the result is simply the number under the root, \( 2 \), because the square root and the square cancel each other out.

• This concept: \( (\sqrt{a} \times \sqrt{a} = a) \), is crucial in solving such problems.

Being comfortable with square roots means one can easily break down and comprehend expressions, allowing for effective handling of equations and expressions in various algebraic scenarios.