To simplify the expression \(\sqrt{\frac{a}{x}} \left( \sqrt{\frac{1}{ax}} \left( \sqrt{\frac{a}{x^3}} \right) \right)\), we need to break it down step by step.
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Start with the innermost square root: \[ \sqrt{\frac{a}{x^3}} = \frac{\sqrt{a}}{\sqrt{x^3}} = \frac{\sqrt{a}}{x^{3/2}} \]
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Substitute this into the next outer square root: \[ \sqrt{\frac{1}{ax}} \left( \frac{\sqrt{a}}{x^{3/2}} \right) \] We can first simplify \(\sqrt{\frac{1}{ax}}\): \[ \sqrt{\frac{1}{ax}} = \frac{1}{\sqrt{a}\sqrt{x}} = \frac{1}{\sqrt{a} x^{1/2}} \] Now combining: \[ \frac{1}{\sqrt{a} x^{1/2}} \cdot \frac{\sqrt{a}}{x^{3/2}} = \frac{\sqrt{a}}{\sqrt{a} x^{1/2} \cdot x^{3/2}} = \frac{1}{x^{2}} \]
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Substitute this result into the remaining square root: \[ \sqrt{\frac{a}{x}} \cdot \frac{1}{x^{2}} = \frac{\sqrt{a}}{\sqrt{x}} \cdot \frac{1}{x^{2}} = \frac{\sqrt{a}}{x^{2} \sqrt{x}} = \frac{\sqrt{a}}{x^{2.5}} = \frac{\sqrt{a}}{x^{5/2}} \]
Putting it all together, the final simplified expression is: \[ \frac{\sqrt{a}}{x^{5/2}} \]
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