Question

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

Answer 1

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:

1. Simplify the innermost expression:
\[
\sqrt{\frac{a}{x^3}} = \frac{\sqrt{a}}{x^{3/2}}
\]
So, substituting this back gives us:
\[
\frac{1}{a x \sqrt{\frac{a}{x^3}}} = \frac{1}{a x \cdot \frac{\sqrt{a}}{x^{3/2}}} = \frac{1}{\frac{a \sqrt{a}}{x^{1/2}}} = \frac{x^{1/2}}{a \sqrt{a}}
\]

2. Now substitute this into the next square root:
\[
\sqrt{\frac{1}{a x \sqrt{\frac{a}{x^3}}}} = \sqrt{\frac{x^{1/2}}{a \sqrt{a}}} = \frac{(x^{1/2})^{1/2}}{(a \sqrt{a})^{1/2}} = \frac{x^{1/4}}{(a^{3/2})}
\]

3. Now substituting back into the outermost expression:
\[
\sqrt{\frac{a}{x \cdot \frac{x^{1/4}}{(a^{3/2})}}} = \sqrt{\frac{a (a^{3/2})}{x \cdot x^{1/4}}} = \sqrt{\frac{a^{5/2}}{x^{5/4}}}
\]

4. 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!