The image contains a math problem where we are asked to find the value of $X$ that satisfies the given expression. Let's extract and solve the equation step-by-step.

Problem breakdown:

The equation in the image is: [ \frac{\sqrt{5} \cdot \sqrt[3]{5^{3}} \cdot \sqrt[5]{5^{6}} \cdot 5^{\frac{1}{2}} \cdot 5^{\frac{2}{3}} \cdot \sqrt[10]{5^{9}}}{5^{X}} = 5 ]

We need to simplify the left-hand side, collect all the exponents of 5, and solve for $X$.

Step 1: Simplify the powers of 5.

Using exponent rules, we can simplify the expression by combining the terms in the numerator:

1. $\sqrt{5} = 5^{\frac{1}{2}}$
2. $\sqrt[3]{5^{3}} = 5^{1}$
3. $\sqrt[5]{5^{6}} = 5^{\frac{6}{5}}$
4. $5^{\frac{1}{2}}$ (remains as is)
5. $5^{\frac{2}{3}}$ (remains as is)
6. $\sqrt[10]{5^{9}} = 5^{\frac{9}{10}}$

Step 2: Add the exponents in the numerator.

Now we sum all the exponents: $\frac{1}{2} + 1 + \frac{6}{5} + \frac{1}{2} + \frac{2}{3} + \frac{9}{10}$

Step 3: Solve for $X$.

Let me compute this and solve for $X$.The value of $X$ that satisfies the equation is approximately $X = 3.7667$.

Would you like more details on this solution, or do you have any other questions?

Here are 5 related questions for further practice:

1. How do you simplify powers with fractional exponents?
2. What are the properties of exponents when multiplying terms with the same base?
3. How do you solve equations involving radicals and fractional exponents?
4. What is the general formula for a radical expression as an exponent?
5. How do exponent rules apply when dealing with roots like square roots and cube roots?

Tip: When simplifying exponent expressions, always combine the terms carefully and remember the basic exponent rules like adding exponents when multiplying terms with the same base.