https://file.aiquickdraw.com/mathbot/file/1727029595271telmfboi.jpg

The equation in the image is:

\[
\frac{\sqrt{x y} \cdot x^{\frac{1}{3}} \cdot 2y^{14}}{\left(x^{10} y^9\right)^{\frac{1}{12}}} = 2
\]

### Step-by-Step Solution:

1. **Simplify the numerator:**
   - \(\sqrt{xy} = (xy)^{\frac{1}{2}} = x^{\frac{1}{2}} y^{\frac{1}{2}}\)
   - We already have \(x^{\frac{1}{3}}\) and \(2y^{14}\), so the numerator becomes:
     \[
     x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot x^{\frac{1}{3}} \cdot 2y^{14}
     \]
   - Combine the powers of \(x\) and \(y\):
     \[
     x^{\frac{1}{2} + \frac{1}{3}} \cdot y^{\frac{1}{2} + 14} \cdot 2
     \]
     To combine the powers of \(x\), find a common denominator:
     \[
     \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}
     \]
     Thus, the numerator becomes:
     \[
     2x^{\frac{5}{6}} y^{\frac{29}{2}}
     \]

2. **Simplify the denominator:**
   - \(\left(x^{10} y^9\right)^{\frac{1}{12}}\) means applying the fractional exponent to both \(x^{10}\) and \(y^9\):
     \[
     \left(x^{10} y^9\right)^{\frac{1}{12}} = x^{\frac{10}{12}} y^{\frac{9}{12}} = x^{\frac{5}{6}} y^{\frac{3}{4}}
     \]

3. **Combine the expression:**
   Now the equation becomes:
   \[
   \frac{2x^{\frac{5}{6}} y^{\frac{29}{2}}}{x^{\frac{5}{6}} y^{\frac{3}{4}}} = 2
   \]
   - Cancel out \(x^{\frac{5}{6}}\) (since it appears in both the numerator and the denominator):
     \[
     \frac{2y^{\frac{29}{2}}}{y^{\frac{3}{4}}} = 2
     \]
   - Simplify the powers of \(y\):
     \[
     y^{\frac{29}{2} - \frac{3}{4}} = y^{\frac{58}{4} - \frac{3}{4}} = y^{\frac{55}{4}}
     \]
     The equation now becomes:
     \[
     2y^{\frac{55}{4}} = 2
     \]

4. **Solve for \(y\):**
   - Divide both sides by 2:
     \[
     y^{\frac{55}{4}} = 1
     \]
   - Take the reciprocal of the exponent to solve for \(y\):
     \[
     y = 1^{\frac{4}{55}} = 1
     \]

So, \(y = 1\).

### Verification:
Substitute \(y = 1\) back into the original equation. Since any power of 1 is still 1, all terms involving \(y\) simplify to 1, leaving the equation to depend only on \(x\), which simplifies correctly. Thus, \(y = 1\) is the correct solution.

Would you like further details on any step?

### Related Questions:
1. What are the properties of fractional exponents and how do they simplify?
2. How do you solve equations involving roots and exponents?
3. Can you provide more examples where powers of variables are combined?
4. How would the solution change if the exponent values were negative?
5. What are other methods to simplify algebraic fractions with exponents?

**Tip:** Always simplify the terms step by step to avoid errors when dealing with fractional exponents.

Solve the equation: (√(xy) * x^(1/3) * 2y^14) / (x^10 y^9)^(1/12) = 2

Learn how to simplify and solve algebraic fractions with fractional exponents using the equation: (√(xy) * x^(1/3) * 2y^14) / (x^10 y^9)^(1/12) = 2. This problem covers key algebraic concepts involving exponent rules, simplification, and solving for unknown variables. Suitable for high school students in Grades 9-11.

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