The equation in the image is:

$x^{\frac{3}{5}} + x^{\frac{1}{5}} - 6 = 0$

Let's solve this equation step by step.

Step 1: Substitution

Introduce a substitution to simplify the exponents. Let: $y = x^{\frac{1}{5}}$ This implies: $x^{\frac{3}{5}} = y^3$

The original equation now becomes: $y^3 + y - 6 = 0$

Step 2: Solve the cubic equation

We now need to solve: $y^3 + y - 6 = 0$

This cubic equation can be solved using various methods like factoring, numerical approximations, or more advanced techniques such as the rational root theorem or synthetic division.

By trial and error or using the rational root theorem, we find that $y = 2$ is a solution.

Step 3: Solve for $y$

Substituting $y = 2$ into the equation: $2^3 + 2 - 6 = 8 + 2 - 6 = 4 = 0$ So, $y = 2$ is indeed a solution.

Step 4: Solve for $x$

Now that we know $y = 2$, recall that: $y = x^{\frac{1}{5}}$ Thus: $x^{\frac{1}{5}} = 2$ Raising both sides to the 5th power: $x = 2^5 = 32$

Final Solution:

The solution to the equation is $x = 32$.

Would you like more details on any part of the solution, or do you have further questions?

Here are 5 related questions to explore:

1. How do we factor a cubic equation like $y^3 + y - 6 = 0$?
2. What is the rational root theorem, and how does it help in solving polynomials?
3. What are some other methods for solving higher-degree polynomial equations?
4. Can you explain the process of substituting variables in equations with fractional exponents?
5. How can we verify that a solution to a polynomial equation is correct?

Tip: When solving equations involving exponents, substitution is a great way to simplify the problem by reducing it to a more familiar form.