https://r2.erweima.ai/mathbot/image/17250185676165wxs4xu0.jpg

resuelveme la operacion y simplifique la respuesta

Let's solve the given mathematical expression step by step.

### Step 1: Analyze and Simplify Each Fraction

The expression given is:

\[
\left[ \left( \frac{\frac{1}{5} - \frac{6}{4}}{\frac{3}{2} + \frac{7}{10}} \right)^{-1} \left( \frac{\frac{1}{2} + \frac{2}{3} - \frac{31}{33}}{\frac{5}{4} + \frac{9}{4} + \frac{\frac{8}{13} - 1}{}} \right) \right]^4
\]

#### 1. Simplify the Inner Fraction on the Left Side:

The left side of the expression inside the brackets is:

\[
\frac{\frac{1}{5} - \frac{6}{4}}{\frac{3}{2} + \frac{7}{10}}
\]

**Simplify the numerator:**

\[
\frac{1}{5} - \frac{6}{4} = \frac{1}{5} - \frac{3}{2}
\]

To subtract these, find a common denominator, which is 10:

\[
\frac{1}{5} = \frac{2}{10}, \quad \frac{3}{2} = \frac{15}{10}
\]

Thus,

\[
\frac{1}{5} - \frac{3}{2} = \frac{2}{10} - \frac{15}{10} = \frac{-13}{10}
\]

**Simplify the denominator:**

\[
\frac{3}{2} + \frac{7}{10}
\]

Find a common denominator, which is 10:

\[
\frac{3}{2} = \frac{15}{10}, \quad \frac{7}{10} = \frac{7}{10}
\]

Thus,

\[
\frac{3}{2} + \frac{7}{10} = \frac{15}{10} + \frac{7}{10} = \frac{22}{10} = \frac{11}{5}
\]

**Combine the fraction:**

\[
\frac{-13}{10} \div \frac{11}{5} = \frac{-13}{10} \times \frac{5}{11} = \frac{-13 \times 5}{10 \times 11} = \frac{-65}{110}
\]

Simplify by dividing the numerator and denominator by 5:

\[
\frac{-65}{110} = \frac{-13}{22}
\]

#### 2. Simplify the Inner Fraction on the Right Side:

The right side inside the brackets is:

\[
\frac{\frac{1}{2} + \frac{2}{3} - \frac{31}{33}}{\frac{5}{4} + \frac{9}{4} + \frac{\frac{8}{13} - 1}{}} 
\]

**Simplify the numerator:**

\[
\frac{1}{2} + \frac{2}{3} - \frac{31}{33}
\]

Find a common denominator for these fractions, which is 66:

\[
\frac{1}{2} = \frac{33}{66}, \quad \frac{2}{3} = \frac{44}{66}, \quad \frac{31}{33} = \frac{62}{66}
\]

Thus,

\[
\frac{1}{2} + \frac{2}{3} - \frac{31}{33} = \frac{33}{66} + \frac{44}{66} - \frac{62}{66} = \frac{15}{66}
\]

Simplify by dividing by 3:

\[
\frac{15}{66} = \frac{5}{22}
\]

**Simplify the denominator:**

\[
\frac{5}{4} + \frac{9}{4} + \left( \frac{8}{13} - 1 \right)
\]

Simplify inside the parenthesis:

\[
\frac{8}{13} - 1 = \frac{8}{13} - \frac{13}{13} = \frac{-5}{13}
\]

Combine the fractions:

\[
\frac{5}{4} + \frac{9}{4} = \frac{14}{4} = \frac{7}{2}
\]

Convert \(\frac{7}{2}\) and \(\frac{-5}{13}\) to a common denominator (26):

\[
\frac{7}{2} = \frac{91}{26}, \quad \frac{-5}{13} = \frac{-10}{26}
\]

Add these fractions:

\[
\frac{91}{26} + \frac{-10}{26} = \frac{81}{26}
\]

**Combine the fraction:**

\[
\frac{\frac{5}{22}}{\frac{81}{26}} = \frac{5}{22} \times \frac{26}{81} = \frac{130}{1782} 
\]

Simplify by dividing by 2:

\[
\frac{130}{1782} = \frac{65}{891}
\]

### Step 3: Combine Everything Together and Solve

Now we have:

\[
\left[ \left(\frac{-13}{22}\right)^{-1} \cdot \frac{65}{891} \right]^4
\]

Simplify the inverse:

\[
\left(-\frac{22}{13} \times \frac{65}{891}\right)^4 = \left(-\frac{22 \times 65}{13 \times 891}\right)^4
\]

Simplify inside the parentheses:

\[
- \frac{1430}{11583}
\]

Now raise this to the power of 4:

\[
\left(-\frac{1430}{11583}\right)^4 = \frac{1430^4}{11583^4}
\]

Finally, simplifying these values:

- \(\frac{1430^4}{11583^4}\) is quite large and does not simplify nicely to an integer. 

Thus, the simplified form of this expression is:

\[
\left(-\frac{1430}{11583}\right)^4
\]

Would you like more details on any step or have any specific questions?

### Further Questions:
1. Would you like to calculate the decimal representation of the result?
2. Are you interested in seeing the multiplication process for the fractions?
3. Would you like to know more about how to find common denominators?
4. Do you need help with the concept of raising fractions to a power?
5. Would you like to explore why certain steps involved finding the inverse of fractions?

**Tip:** Always look for common denominators when adding or subtracting fractions to simplify your calculations.

Explore a step-by-step solution to simplify and solve a complex mathematical expression involving fractions and exponentiation. This problem covers advanced concepts in fractions, inverse operations, and exponentiation, suitable for advanced mathematics students.

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