The expression you provided appears to be nested and contains some operations that may be misformatted. Let's clarify and simplify it step by step.
The expression seems to be:
\[ \sqrt{\frac{a}{x \sqrt{\frac{1}{a x \sqrt{\frac{a}{x^3}}}}}} \]
To simplify this, we'll break it down:
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Simplify the innermost expression: \[ \sqrt{\frac{a}{x^3}} = \frac{\sqrt{a}}{x^{3/2}} \] So, substituting this back gives us: \[ \frac{1}{a x \sqrt{\frac{a}{x^3}}} = \frac{1}{a x \cdot \frac{\sqrt{a}}{x^{3/2}}} = \frac{1}{\frac{a \sqrt{a}}{x^{1/2}}} = \frac{x^{1/2}}{a \sqrt{a}} \]
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Now substitute this into the next square root: \[ \sqrt{\frac{1}{a x \sqrt{\frac{a}{x^3}}}} = \sqrt{\frac{x^{1/2}}{a \sqrt{a}}} = \frac{(x^{1/2})^{1/2}}{(a \sqrt{a})^{1/2}} = \frac{x^{1/4}}{(a^{3/2})} \]
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Now substituting back into the outermost expression: \[ \sqrt{\frac{a}{x \cdot \frac{x^{1/4}}{(a^{3/2})}}} = \sqrt{\frac{a (a^{3/2})}{x \cdot x^{1/4}}} = \sqrt{\frac{a^{5/2}}{x^{5/4}}} \]
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Finally, simplifying this gives us: \[ \sqrt{\frac{a^{5/2}}{x^{5/4}}} = \frac{a^{5/4}}{x^{5/8}} \]
Thus, the simplified form of the original expression is:
\[ \frac{a^{5/4}}{x^{5/8}} \]
Note: If your original expression was different due to formatting issues, please correct it and I'll be happy to help!
1 answer