Question

sqrt(a/x)(sqrt(1/ax)(sqrt(a/x^3))))

Answer 1

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.

1. Start with the innermost square root:
\[
\sqrt{\frac{a}{x^3}} = \frac{\sqrt{a}}{\sqrt{x^3}} = \frac{\sqrt{a}}{x^{3/2}}
\]

2. 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:
\[
\frac{1}{\sqrt{a} x^{1/2}} \cdot \frac{\sqrt{a}}{x^{3/2}} = \frac{\sqrt{a}}{\sqrt{a} x^{1/2} \cdot x^{3/2}} = \frac{1}{x^{2}}
\]

3. 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:
\[
\frac{\sqrt{a}}{x^{5/2}}
\]