QQuestionMathematics
QuestionMathematics
Find the simplified product where $x \geq 0: \sqrt{5 x}\left(\sqrt{8 x^{2}}- 2 \sqrt{x}\right)$
- $\sqrt{10 x}$
- $2 x \sqrt{40 x}- 2 x$
- $2 x \sqrt{10 x}- 2 \sqrt{5 x}$
- $2 x \sqrt{10 x}- 2 x \sqrt{5}$
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Answer
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Step 1:: Distribute the \sqrt{5x} inside the parentheses, being careful with the signs.
\sqrt{5x}\left(\sqrt{8x^2} - 2\sqrt{x}\right) = \sqrt{5x}\sqrt{8x^2} - \sqrt{5x} \cdot 2\sqrt{x}
Step 2:: Simplify the terms on the right side of the equation.
Recall the product rule for radicals: \sqrt{a}\sqrt{b} = \sqrt{ab} \begin{align*} \sqrt{5x}\sqrt{8x^2} - \sqrt{5x} \cdot 2\sqrt{x} &= \sqrt{5x \cdot 8x^2} - 2\sqrt{5x^2} \ &= \sqrt{40x^3} - 2x\sqrt{5} \end{align*}
Step 3:: Simplify the first term further by taking out an x factor.
\sqrt{40x^3} = \sqrt{40x \cdot x^2} = \sqrt{40x} \cdot x = 2x\sqrt{10x}
Step 4:: Substitute this back into the equation to get the simplified product.
\sqrt{40x^3} - 2x\sqrt{5} = 2x\sqrt{10x} - 2x\sqrt{5}
Final Answer
The simplified product is 2x\sqrt{10x} - 2x\sqrt{5}.
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