To simplify $\frac{20\sqrt{54}}{5\sqrt{2}}$:

Step 1: Simplify the coefficients

The coefficients $20$ and $5$ can be simplified: $\frac{20}{5} = 4$ This leaves: $4 \cdot \frac{\sqrt{54}}{\sqrt{2}}$

Step 2: Combine the square roots

Use the property $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$: $\frac{\sqrt{54}}{\sqrt{2}} = \sqrt{\frac{54}{2}} = \sqrt{27}$

Step 3: Simplify $\sqrt{27}$

The number $27$ can be written as $9 \cdot 3$, and $\sqrt{9} = 3$: $\sqrt{27} = \sqrt{9 \cdot 3} = 3\sqrt{3}$

Step 4: Substitute back

Now substitute $\sqrt{27} = 3\sqrt{3}$ into the expression: $4 \cdot 3\sqrt{3} = 12\sqrt{3}$

Final Answer:

$\frac{20\sqrt{54}}{5\sqrt{2}} = 12\sqrt{3}$

Would you like more details on any step?

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Related Questions:

1. How do you simplify other expressions with radicals like $\sqrt{48}/\sqrt{3}$?
2. What are the rules for multiplying and dividing square roots?
3. How do you rationalize a denominator in radical expressions?
4. Can you explain how to simplify $\sqrt{50} + \sqrt{18}$?
5. What are some practical uses of radicals in geometry?

Tip:

Always check if a square root can be simplified by identifying perfect squares in its factors!