The given expression from the image is:

$\frac{\sqrt{54a^3}}{\sqrt{2a}}$

Step 1: Simplify each square root

Using the property of square roots: $\sqrt{\frac{x}{y}} = \frac{\sqrt{x}}{\sqrt{y}},$ we can rewrite the entire square root as a single square root: $\frac{\sqrt{54a^3}}{\sqrt{2a}} = \sqrt{\frac{54a^3}{2a}}.$

Step 2: Simplify the fraction under the square root

Simplify $\frac{54a^3}{2a}$: $\frac{54a^3}{2a} = \frac{54}{2} \cdot \frac{a^3}{a} = 27a^2.$

So, the expression becomes: $\sqrt{27a^2}.$

Step 3: Simplify the square root

Split the square root using the property $\sqrt{xy} = \sqrt{x} \cdot \sqrt{y}$: $\sqrt{27a^2} = \sqrt{27} \cdot \sqrt{a^2}.$

• The square root of $a^2$ is $a$.
• Simplify $\sqrt{27}$: $\sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3}.$

Thus, $\sqrt{27a^2} = 3a\sqrt{3}$.

Final Answer:

$\frac{\sqrt{54a^3}}{\sqrt{2a}} = 3a\sqrt{3}.$

Let me know if you'd like further clarifications or have any questions!

Related Questions:

1. How can we simplify a square root containing both constants and variables?
2. What are the key properties of square roots used in simplifying radical expressions?
3. How do we handle division of square roots with different bases?
4. Can $\sqrt{54a^3}$ be simplified before dividing by $\sqrt{2a}$?
5. How do we verify the result of such simplifications algebraically?

Tip:

Always check for factors that are perfect squares when simplifying square roots to ensure complete simplification!