https://file.aiquickdraw.com/mathbot/file/1726748502742avlkx78v.jpg

The equation given in the image is:

\[
\sqrt{\frac{a}{b}} = \left(\frac{b}{a}\right)^{1 - 3x}
\]

We need to solve for \( x \).

### Step-by-step solution:

1. **Square both sides**: 
   To eliminate the square root on the left-hand side, square both sides of the equation:

   \[
   \left(\sqrt{\frac{a}{b}}\right)^2 = \left(\left(\frac{b}{a}\right)^{1 - 3x}\right)^2
   \]
   Simplifying:
   \[
   \frac{a}{b} = \left(\frac{b}{a}\right)^{2(1 - 3x)}
   \]
   \[
   \frac{a}{b} = \left(\frac{b}{a}\right)^{2 - 6x}
   \]

2. **Rewrite the right side**:
   Use the property of exponents: 
   \[
   \left(\frac{b}{a}\right)^{2 - 6x} = \left(\frac{a}{b}\right)^{-(2 - 6x)}
   \]
   Now the equation becomes:
   \[
   \frac{a}{b} = \left(\frac{a}{b}\right)^{-(2 - 6x)}
   \]

3. **Set the exponents equal**:
   Since the bases on both sides are the same, we can set the exponents equal to each other:
   \[
   1 = -(2 - 6x)
   \]
   Simplifying:
   \[
   1 = -2 + 6x
   \]

4. **Solve for \( x \)**:
   Add 2 to both sides:
   \[
   3 = 6x
   \]
   Divide by 6:
   \[
   x = \frac{3}{6} = \frac{1}{2}
   \]

Thus, the solution is:
\[
x = \frac{1}{2}
\]

Let me know if you would like further clarification or details! 

### Related questions:
1. What are the properties of exponents used in this problem?
2. How do you deal with fractional exponents in equations?
3. Can this equation be solved using logarithms? Why or why not?
4. How do you check if a solution for \( x \) is valid by substituting it back into the original equation?
5. What other methods could be used to simplify the square root?

**Tip**: Always remember to check for extraneous solutions when dealing with square roots and powers.

Explore how to solve the exponential equation sqrt(a/b) = (b/a)^(1 - 3x) with a detailed step-by-step guide. This problem demonstrates the use of exponent rules and solving techniques for equations. Suitable for high school students studying algebra.

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