To simplify the expression $\sqrt{54x^7}$, we will first break it down into parts that include perfect squares and parts that do not.

1. Factorize 54:
   We begin by factorizing the constant $54$. $54 = 2 \times 3^3$ So we have: $\sqrt{54x^7} = \sqrt{2 \times 3^3 \times x^7}$

2. Extract perfect squares:
   We now look for perfect squares inside the square root.

   • $3^3$ can be rewritten as $3^2 \times 3$, where $3^2$ is a perfect square.
   • $x^7$ can be rewritten as $x^6 \times x$, where $x^6$ is a perfect square.

   So, we now have: $\sqrt{54x^7} = \sqrt{2 \times 3^2 \times 3 \times x^6 \times x}$

3. Simplify the square root:
   We can now pull out the perfect squares: $\sqrt{54x^7} = \sqrt{(3^2 \times x^6) \times 2 \times 3 \times x}$ $= 3x^3 \sqrt{6x}$

Thus, the simplified form of $\sqrt{54x^7}$ is: $3x^3 \sqrt{6x}$

Do you need further explanation or details on any step?

Here are some related questions you might find interesting:

1. How would you simplify $\sqrt{72y^8}$?
2. What is the process for simplifying square roots of polynomials?
3. How do you identify perfect squares when working with variables?
4. What happens if the variable inside the square root is negative?
5. How can you simplify a square root with a rational expression?

Tip: When simplifying square roots, always break down constants and variables into their prime factors and pull out any perfect squares!