To simplify the expression
\[ \sqrt{\frac{a}{x}\sqrt{\frac{1}{ax}\frac{\sqrt{a}}{\sqrt{x^{3}}}}} \]
we'll break it down step by step.
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Inside the inner square root:
Let's first simplify the expression inside the inner square root: \[ \frac{1}{ax} \cdot \frac{\sqrt{a}}{\sqrt{x^3}} = \frac{\sqrt{a}}{ax\sqrt{x^3}} = \frac{\sqrt{a}}{a\sqrt{x}\cdot x^{3/2}} = \frac{\sqrt{a}}{a x^{2}} = \frac{1}{\sqrt{a}} \cdot \frac{1}{x^{2}} \]
So the inner square root simplifies to: \[ \sqrt{\frac{1}{ax}\cdot \frac{\sqrt{a}}{\sqrt{x^3}}} = \sqrt{\frac{1}{a x^2}} \]
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Now substitute back:
Substitute back into the outer square root: \[ \sqrt{\frac{a}{x} \cdot \sqrt{\frac{1}{a x^2}}} \]
This becomes: \[ \sqrt{\frac{a}{x} \cdot \frac{1}{\sqrt{a x^2}}} = \sqrt{\frac{a}{x} \cdot \frac{1}{\sqrt{a} \sqrt{x^2}}} = \sqrt{\frac{a}{x} \cdot \frac{1}{\sqrt{a}x}} = \sqrt{\frac{1}{\sqrt{a}} \cdot \frac{1}{x^2}} \]
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Combine fractions:
Combine the components: \[ \sqrt{\frac{1}{\sqrt{a} \cdot x^2}} = \frac{1}{\sqrt{\sqrt{a} \cdot x^2}} = \frac{1}{\sqrt{a^{1/2} x^2}} = \frac{1}{\sqrt{a^{1/2}} \cdot \sqrt{x^2}} = \frac{1}{\sqrt{a^{1/2}} \cdot x} = \frac{1}{\sqrt{a} x} \]
So the final simplified form of the original expression is
\[ \frac{1}{\sqrt{a} x} \]
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