https://r2.erweima.ai/mathbot/image/1724117083980gzo4pkge.jpg

analiza la imagen y encuentra exponente final  de x

The image appears to be asking for the simplification of a mathematical expression involving exponents.

The expression given in the image is:

\[
\left(\frac{x^4}{y^3} \right)^{\frac{3}{5}} \times \left(\frac{y^6}{x^2}\right)^{\frac{5}{6}}
\]

To find the final exponent of \( x \), we'll simplify the expression step by step.

### Step 1: Simplify Each Term Separately

1. **First term:**  
   \[
   \left(\frac{x^4}{y^3} \right)^{\frac{3}{5}} = \frac{x^{4 \cdot \frac{3}{5}}}{y^{3 \cdot \frac{3}{5}}} = \frac{x^{\frac{12}{5}}}{y^{\frac{9}{5}}}
   \]

2. **Second term:**  
   \[
   \left(\frac{y^6}{x^2}\right)^{\frac{5}{6}} = \frac{y^{6 \cdot \frac{5}{6}}}{x^{2 \cdot \frac{5}{6}}} = \frac{y^5}{x^{\frac{5}{3}}}
   \]

### Step 2: Multiply the Simplified Terms

Now, multiply the two simplified terms:

\[
\frac{x^{\frac{12}{5}}}{y^{\frac{9}{5}}} \times \frac{y^5}{x^{\frac{5}{3}}}
\]

Combine the exponents of \( x \) and \( y \) separately:

\[
x^{\frac{12}{5} - \frac{5}{3}} \times y^{5 - \frac{9}{5}}
\]

### Step 3: Simplify the Exponents

1. **For \( x \):**
   \[
   \frac{12}{5} - \frac{5}{3} = \frac{36}{15} - \frac{25}{15} = \frac{11}{15}
   \]
   So, the exponent of \( x \) is \( \frac{11}{15} \).

2. **For \( y \):**
   \[
   5 - \frac{9}{5} = \frac{25}{5} - \frac{9}{5} = \frac{16}{5}
   \]

### Final Expression

Thus, the final expression is:

\[
x^{\frac{11}{15}} \times y^{\frac{16}{5}}
\]

The final exponent of \( x \) is \( \frac{11}{15} \).

Would you like further details or have any other questions?

Here are some related questions for further practice:

1. How would you simplify the expression if the exponents had been negative?
2. What would be the exponent of \( y \) in the final expression?
3. How do the rules of exponents change when dealing with roots instead of powers?
4. What if the original expression involved division instead of multiplication?
5. How would the problem change if the expression included a constant term?

**Tip:** When simplifying expressions involving exponents, always perform operations step by step to avoid mistakes.

Learn how to simplify the given exponential expression involving fractions and exponents. Follow a step-by-step solution to find the final exponent of x and understand the rules of exponents applied in the process. Suitable for high school students.

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