Problem 3 - Intermediate Algebra

Question

Add or subtract. $$ 2 \sqrt{2 x^{3}}+4 x \sqrt{8 x} $$

Step-by-Step Solution

Answer

The simplified expression is \(10x^{3/2}\sqrt{2}\).

Step 1: Simplifying the Radicals

The expression is initially given as \(2 \sqrt{2 x^{3}} + 4 x \sqrt{8 x}\). First, simplify each radical term individually. \(\sqrt{2 x^3}\) simplifies as \(x^{3/2} \sqrt{2}\) since \(x^{3} = (x^{2} \cdot x) = x^{2} \cdot x\), and \(x^2\) can be taken out of the square root as \(x\). So, we have \(2x^{3/2} \sqrt{2}\). For \(\sqrt{8x}\), since \(8 = 4 \cdot 2\) and \(\sqrt{4} = 2\), it simplifies to \(2 \sqrt{2x}\).

Step 2: Substitute Simplified Terms Back

Return to the entire expression with these simplifications: \(2x^{3/2} \sqrt{2} + 4x (2\sqrt{2x})\). Distribute the \(4x\) term in the second part: \(8x \sqrt{2x}\).

Step 3: Combine Like Terms

Observe the structure of the expression: \(2x^{3/2}\sqrt{2} + 8x \sqrt{2} \cdot \sqrt{x}\). Since both terms involve a factor of \(\sqrt{2}\), write it as \(2x^{3/2} \sqrt{2} + 8x^{3/2} \sqrt{2}\) (noting that \(\sqrt{x}\) becomes \(x^{1/2}\)). Factor out \(\sqrt{2}\) to obtain: \((2x^{3/2} + 8x^{3/2})\sqrt{2}\).

Step 4: Factor Before Final Combination

Upon factoring \(x^{3/2}\sqrt{2}\), the expression becomes \((2 + 8) x^{3/2} \sqrt{2}\). This simplifies final to \(10x^{3/2}\sqrt{2}\).

Key Concepts

Simplifying Radicals

When we talk about simplifying radicals, it's all about breaking them down into their most basic parts. Let's take a look at the radicals in our expression: \(2 \sqrt{2 x^{3}}\) and \(4 x \sqrt{8 x}\). The goal is to express each radical in a simpler form. For \(\sqrt{2 x^{3}}\), we notice that it's the same as \(\sqrt{x^2 \cdot x}\), allowing us to take \(x\) out of the square root and leaving us with \(x^{1.5} \sqrt{2}\). Similarly, for \(\sqrt{8x}\), since 8 is \(4\cdot 2\), and \(\sqrt{4} = 2\), it simplifies to \(2 \sqrt{2x}\). Remember:

• Find factors inside the radical which are perfect squares.
• Take out the square root of those factors.
• Simplify step-by-step for clarity.

Understanding this process will make dealing with radicals much easier in any algebraic context.

Combining Like Terms

This concept refers to rearranging our expression so that terms with the same radical or variable parts are grouped together. In the simplified expression from the previous section, we had terms like \(2x^{1.5} \sqrt{2}\) and \(8x^{1.5} \sqrt{2}\). Both contain \(x^{1.5} \sqrt{2}\), which makes them like terms.

• Identify components that are identical in form.
• Add or subtract these like terms just as you would combine similar numerical terms.

This step showcases how understanding the structure of expressions helps make algebra less daunting and certainly more methodical. In this way, by combining, we move toward a simpler, more concise expression.

Expression Simplification

Expression simplification is straightforward if you follow each step diligently. Once the radicals and variables are simplified, and like terms are identified, it's about combining everything neatly. From our already clear terms \(2x^{1.5} \sqrt{2}\) and \(8x^{1.5} \sqrt{2}\), we sum them to get \((2 + 8)x^{1.5} \sqrt{2}\).

• Breaking expressions down into simpler parts.
• Combining all parts for a neater representation.
• Avoiding unnecessary complexity by removing redundant factors.

This approach not only helps simplify but also results in an expression that's quick to interpret, easily demonstrated as \(10x^{1.5}\sqrt{2}\). Such simplification is ultimately what streamlines solving algebraic problems.

Factoring Expressions

Factoring can feel like solving a puzzle with algebraic expressions. It requires seeing common elements and "pulling them out" to simplify your work. In our exercise, after simplifying and combining, \(x^{1.5}\sqrt{2}\) is already factored out, revealing the core operation: adding 2 and 8. This gives us a factored expression \((2 + 8) x^{1.5} \sqrt{2}\).

• Look for common factors within terms you can factor out.
• Use factoring to simplify expressions and reduce steps in calculations.
• Always check your factored results to ensure they replace original expressions accurately.

Factoring not only makes calculations straightforward but also enhances clarity and efficiency in problem-solving. Practicing these tips leads to a better grasp of algebra fundamentals.