https://r2.erweima.ai/mathbot/image/1725640240056wdsi4cad.jpg

The expression you uploaded is quite complex, but let's break it down for clarity. Here's the expression as I interpret it:

\[
\left( \sqrt[5]{\frac{b^{2} c^{-1/2}}{b^{-1/2} c^{2}}} \cdot \sqrt[3]{\frac{b^{-1/2} c}{c^{-1/2} b}} \right) - 10
\]

### Steps to Simplify the Expression:
1. **Simplify the terms inside the roots**:
   - For the first root:
   \[
   \frac{b^{2} c^{-1/2}}{b^{-1/2} c^{2}} = b^{2 - (-1/2)} \cdot c^{-1/2 - 2} = b^{2 + 1/2} \cdot c^{-5/2} = b^{5/2} \cdot c^{-5/2}
   \]
   - For the second root:
   \[
   \frac{b^{-1/2} c}{c^{-1/2} b} = b^{-1/2 - 1} \cdot c^{1 - (-1/2)} = b^{-3/2} \cdot c^{3/2}
   \]

2. **Apply the roots**:
   - The first root becomes:
   \[
   \sqrt[5]{b^{5/2} \cdot c^{-5/2}} = \left(b^{5/2} \cdot c^{-5/2}\right)^{1/5} = b^{1/2} \cdot c^{-1/2}
   \]
   - The second root becomes:
   \[
   \sqrt[3]{b^{-3/2} \cdot c^{3/2}} = \left(b^{-3/2} \cdot c^{3/2}\right)^{1/3} = b^{-1/2} \cdot c^{1/2}
   \]

3. **Multiply the results of both roots**:
   \[
   (b^{1/2} \cdot c^{-1/2}) \cdot (b^{-1/2} \cdot c^{1/2}) = b^{1/2 - 1/2} \cdot c^{-1/2 + 1/2} = b^{0} \cdot c^{0} = 1
   \]

4. **Subtract 10**:
   \[
   1 - 10 = -9
   \]

### Final Answer:
The simplified value of the given expression is:

\[
\boxed{-9}
\]

Would you like further details on the simplification process or any step that needs more clarification?

Here are some related questions to explore:
1. How do fractional exponents work in expressions?
2. What are the properties of cube roots and fifth roots?
3. How does simplifying expressions with variables in both the numerator and denominator work?
4. Why do exponents follow addition and subtraction rules when dividing?
5. How can radicals be simplified using prime factorization?

**Tip:** When dealing with roots and exponents, remember that fractional exponents provide a convenient way to handle roots algebraically!

Simplify the expression: ((5th root of (b^2 * c^(-1/2)) / (b^(-1/2) * c^2)) * (3rd root of (b^(-1/2) * c) / (c^(-1/2) * b))) - 10

Learn how to simplify a complex expression involving multiple radicals and exponents. This problem walks through step-by-step solutions, covering key algebraic concepts such as exponentiation, simplifying radicals, and the rules of exponents. Suitable for students in Grades 9-12.

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