Question

what is the following simplified product? assume \\(x \ge 0\\).

\\(\left( \sqrt{6x^2} + 4\sqrt{8x^3} \
ight) \left( \sqrt{9x} - x\sqrt{5x^5} \
ight)\\)

• \\(3x\sqrt{6x} + x^4\sqrt{30x} + 24x^2\sqrt{2x} + 8x^5\sqrt{10x}\\)
• \\(3x\sqrt{6x} + x^4\sqrt{30x} + 24x^2\sqrt{2} + 8x^5\sqrt{10}\\)
• \\(3x\sqrt{6x} - x^4\sqrt{30x} + 24x^2\sqrt{2} - 8x^5\sqrt{10}\\)
• \\(3x\sqrt{6x} - x^4\sqrt{30x} + 24x^2\sqrt{2x} - 8x^5\sqrt{10x}\\)

Explanation:

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<reasoning>

Simplify each radical term

Using the Radical Simplification knowledge point
\[

$$\begin{aligned} \sqrt{6x^2} &= x\sqrt{6} \quad (\text{since } x \ge 0) \\ 4\sqrt{8x^3} &= 4 \cdot 2x\sqrt{2x} = 8x\sqrt{2x} \\ \sqrt{9x} &= 3\sqrt{x} \\ x\sqrt{5x^5} &= x \cdot x^2\sqrt{5x} = x^3\sqrt{5x} \end{aligned}$$

\]

Rewrite the product with simplified terms

Using the Multiplying Binomial Radicals knowledge point
\[
(x\sqrt{6} + 8x\sqrt{2x})(3\sqrt{x} - x^3\sqrt{5x})
\]

Expand the product using FOIL

Using the Simplifying Radical Products knowledge point
\[

$$\begin{aligned} \text{First: } & (x\sqrt{6})(3\sqrt{x}) = 3x\sqrt{6x} \\ \text{Outer: } & (x\sqrt{6})(-x^3\sqrt{5x}) = -x^4\sqrt{30x} \\ \text{Inner: } & (8x\sqrt{2x})(3\sqrt{x}) = 24x\sqrt{2x^2} = 24x^2\sqrt{2} \\ \text{Last: } & (8x\sqrt{2x})(-x^3\sqrt{5x}) = -8x^4\sqrt{10x^2} = -8x^5\sqrt{10} \end{aligned}$$

\]

Combine the expanded terms

Using the Simplifying Radical Products knowledge point
\[
3x\sqrt{6x} - x^4\sqrt{30x} + 24x^2\sqrt{2} - 8x^5\sqrt{10}
\]
</reasoning>

<answer>
<mcq-option>(A) \(3x\sqrt{6x} + x^4\sqrt{30x} + 24x^2\sqrt{2x} + 8x^5\sqrt{10x}\)</mcq-option>
<mcq-option>(B) \(3x\sqrt{6x} + x^4\sqrt{30x} + 24x^2\sqrt{2} + 8x^5\sqrt{10}\)</mcq-option>
<mcq-correct>(C) \(3x\sqrt{6x} - x^4\sqrt{30x} + 24x^2\sqrt{2} - 8x^5\sqrt{10}\)</mcq-correct>
<mcq-option>(D) \(3x\sqrt{6x} - x^4\sqrt{30x} + 24x^2\sqrt{2x} - 8x^5\sqrt{10x}\)</mcq-option>
</answer>

<post_analysis>
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Answer:

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</pre_analysis>

<reasoning>

Simplify each radical term

Using the Radical Simplification knowledge point
\[

$$\begin{aligned} \sqrt{6x^2} &= x\sqrt{6} \quad (\text{since } x \ge 0) \\ 4\sqrt{8x^3} &= 4 \cdot 2x\sqrt{2x} = 8x\sqrt{2x} \\ \sqrt{9x} &= 3\sqrt{x} \\ x\sqrt{5x^5} &= x \cdot x^2\sqrt{5x} = x^3\sqrt{5x} \end{aligned}$$

\]

Rewrite the product with simplified terms

Using the Multiplying Binomial Radicals knowledge point
\[
(x\sqrt{6} + 8x\sqrt{2x})(3\sqrt{x} - x^3\sqrt{5x})
\]

Expand the product using FOIL

Using the Simplifying Radical Products knowledge point
\[

$$\begin{aligned} \text{First: } & (x\sqrt{6})(3\sqrt{x}) = 3x\sqrt{6x} \\ \text{Outer: } & (x\sqrt{6})(-x^3\sqrt{5x}) = -x^4\sqrt{30x} \\ \text{Inner: } & (8x\sqrt{2x})(3\sqrt{x}) = 24x\sqrt{2x^2} = 24x^2\sqrt{2} \\ \text{Last: } & (8x\sqrt{2x})(-x^3\sqrt{5x}) = -8x^4\sqrt{10x^2} = -8x^5\sqrt{10} \end{aligned}$$

\]

Combine the expanded terms

Using the Simplifying Radical Products knowledge point
\[
3x\sqrt{6x} - x^4\sqrt{30x} + 24x^2\sqrt{2} - 8x^5\sqrt{10}
\]
</reasoning>

<answer>
<mcq-option>(A) \(3x\sqrt{6x} + x^4\sqrt{30x} + 24x^2\sqrt{2x} + 8x^5\sqrt{10x}\)</mcq-option>
<mcq-option>(B) \(3x\sqrt{6x} + x^4\sqrt{30x} + 24x^2\sqrt{2} + 8x^5\sqrt{10}\)</mcq-option>
<mcq-correct>(C) \(3x\sqrt{6x} - x^4\sqrt{30x} + 24x^2\sqrt{2} - 8x^5\sqrt{10}\)</mcq-correct>
<mcq-option>(D) \(3x\sqrt{6x} - x^4\sqrt{30x} + 24x^2\sqrt{2x} - 8x^5\sqrt{10x}\)</mcq-option>
</answer>

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