The expression provided in the image is:

$\frac{\left(x^5\right)^2 \cdot \left(\sqrt{x}\right)^{10}}{\sqrt[3]{x^{21}} \cdot x^{-15}}$

Let's simplify this step by step.

Step 1: Simplify each component

1. Simplify $\left(x^5\right)^2$: $\left(x^5\right)^2 = x^{5 \times 2} = x^{10}$

2. Simplify $\left(\sqrt{x}\right)^{10}$: $\sqrt{x} = x^{1/2}, \quad \text{so} \quad \left(\sqrt{x}\right)^{10} = \left(x^{1/2}\right)^{10} = x^{\frac{10}{2}} = x^5$

3. Simplify $\sqrt[3]{x^{21}}$: $\sqrt[3]{x^{21}} = x^{\frac{21}{3}} = x^7$

4. Simplify $x^{-15}$:
   This term is already simplified.

Step 2: Substitute these into the original expression

Substituting back into the original expression, we get:

$\frac{x^{10} \cdot x^5}{x^7 \cdot x^{-15}}$

Step 3: Combine like terms using the laws of exponents

1. Simplify the numerator: $x^{10} \cdot x^5 = x^{10+5} = x^{15}$

2. Simplify the denominator: $x^7 \cdot x^{-15} = x^{7+(-15)} = x^{-8}$

Now, we have:

$\frac{x^{15}}{x^{-8}}$

Step 4: Simplify the final expression

Using the law of exponents $\frac{x^a}{x^b} = x^{a-b}$:

$\frac{x^{15}}{x^{-8}} = x^{15 - (-8)} = x^{15 + 8} = x^{23}$

Final Answer

The simplified expression is:

$\boxed{x^{23}}$

Would you like further details on any part of this solution?

Here are some related questions you might find helpful:

1. How do you simplify expressions involving roots and exponents?
2. What are the rules for multiplying and dividing expressions with exponents?
3. How does the cube root affect exponents in an expression?
4. What are the steps to simplify a complex fraction with variables?
5. How do negative exponents affect the simplification process?

Tip: Always apply the laws of exponents carefully, especially when dealing with fractional and negative exponents, as they can drastically change the value of the expression.