https://r2.erweima.ai/mathbot/image/1726801900141bjfsjxd8.jpg

The given expression from the image is as follows:

\[
\frac{\sqrt[3]{\frac{5}{a^7}} \cdot \sqrt{\frac{5}{a}}}{\sqrt[5]{\frac{2}{a^3}} \cdot \sqrt[4]{\frac{2}{a^5}}}
\]

Let’s break it down and simplify the expression step by step.

### Numerator:
1. \(\sqrt[3]{\frac{5}{a^7}}\) can be written as:
   \[
   \frac{5^{1/3}}{a^{7/3}}
   \]

2. \(\sqrt{\frac{5}{a}}\) can be written as:
   \[
   \frac{5^{1/2}}{a^{1/2}}
   \]

So, the entire numerator becomes:
\[
\frac{5^{1/3}}{a^{7/3}} \cdot \frac{5^{1/2}}{a^{1/2}} = \frac{5^{1/3 + 1/2}}{a^{7/3 + 1/2}}
\]

Now, adding the exponents of 5 and \(a\):
- \(1/3 + 1/2 = \frac{2}{6} + \frac{3}{6} = \frac{5}{6}\)
- \(7/3 + 1/2 = \frac{14}{6} + \frac{3}{6} = \frac{17}{6}\)

Thus, the numerator becomes:
\[
\frac{5^{5/6}}{a^{17/6}}
\]

### Denominator:
1. \(\sqrt[5]{\frac{2}{a^3}}\) can be written as:
   \[
   \frac{2^{1/5}}{a^{3/5}}
   \]

2. \(\sqrt[4]{\frac{2}{a^5}}\) can be written as:
   \[
   \frac{2^{1/4}}{a^{5/4}}
   \]

So, the entire denominator becomes:
\[
\frac{2^{1/5}}{a^{3/5}} \cdot \frac{2^{1/4}}{a^{5/4}} = \frac{2^{1/5 + 1/4}}{a^{3/5 + 5/4}}
\]

Now, adding the exponents of 2 and \(a\):
- \(1/5 + 1/4 = \frac{4}{20} + \frac{5}{20} = \frac{9}{20}\)
- \(3/5 + 5/4 = \frac{12}{20} + \frac{25}{20} = \frac{37}{20}\)

Thus, the denominator becomes:
\[
\frac{2^{9/20}}{a^{37/20}}
\]

### Putting everything together:
The simplified form of the original expression is:
\[
\frac{\frac{5^{5/6}}{a^{17/6}}}{\frac{2^{9/20}}{a^{37/20}}} = \frac{5^{5/6} \cdot a^{37/20}}{2^{9/20} \cdot a^{17/6}}
\]

Simplifying further:
- Combine powers of \(a\): \(a^{37/20 - 17/6} = a^{37/20 - \frac{170}{60}} = a^{-59/60}\)
-

Simplify the expression involving radical terms from the provided image.

This example walks through the step-by-step simplification of a complex expression involving radicals and exponents. Explore the application of fractional exponents and radical properties, suitable for high school students learning advanced algebra.

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