The expression in the image is:

$y \sqrt{54x^3 y^3}$

Let’s simplify this step by step.

1. Break down the square root term $\sqrt{54x^3 y^3}$. We can split this into:

$\sqrt{54x^3 y^3} = \sqrt{54} \cdot \sqrt{x^3} \cdot \sqrt{y^3}$

2. Simplify each part:

   • $\sqrt{54}$: The prime factorization of 54 is $54 = 2 \cdot 3^3$. So,

$\sqrt{54} = \sqrt{2 \cdot 3^3} = 3 \sqrt{6}$

• $\sqrt{x^3} = x^{3/2}$. This can be written as $x^{1} \cdot \sqrt{x}$, which simplifies to $x\sqrt{x}$.

• $\sqrt{y^3} = y^{3/2}$, which simplifies to $y\sqrt{y}$.

3. Now, substitute these back into the expression:

$y \sqrt{54x^3 y^3} = y \cdot 3 \sqrt{6} \cdot x\sqrt{x} \cdot y\sqrt{y}$

4. Simplify further:

$= 3yx \cdot \sqrt{6xy}$

Thus, the simplified expression is:

$3yx \sqrt{6xy}$

Would you like further details or have any questions?

Here are 5 related questions:

1. How do you simplify expressions with square roots involving variables and constants?
2. What is the general method for simplifying powers of variables under square roots?
3. How do you handle square roots of larger numbers like 54 in different contexts?
4. Can this expression be simplified further under different conditions, such as specific values for $x$ and $y$?
5. How can you use this simplification technique in more complex algebraic equations?

Tip: Always try to factor out perfect squares when simplifying square roots, as it makes the process faster and cleaner.