Let's simplify the given expression:

Expression:

$\frac{\sqrt{3} \cdot \sqrt{57}}{\sqrt{3} - 10}$

Step-by-step Solution:

1. Simplify the numerator:

   • Recall the property of square roots: $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$.
   • Here, $\sqrt{3} \cdot \sqrt{57} = \sqrt{3 \cdot 57} = \sqrt{171}$.
   • So, the numerator becomes $\sqrt{171}$.

2. Expression now becomes: $\frac{\sqrt{171}}{\sqrt{3} - 10}.$

3. Simplify $\sqrt{171}$:

   • Factorize $171$: $171 = 3 \cdot 57 = 3 \cdot 3 \cdot 19 = 3^2 \cdot 19$.
   • Take the square root of $3^2$: $\sqrt{3^2 \cdot 19} = 3\sqrt{19}$.
   • So, $\sqrt{171} = 3\sqrt{19}$.

4. Substitute back into the expression: $\frac{3\sqrt{19}}{\sqrt{3} - 10}.$

This is the simplified expression.

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If further simplification is needed (e.g., rationalizing the denominator), let me know! 😊

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Questions for Practice:

1. Simplify $\frac{\sqrt{45}}{\sqrt{5} + 3}$.
2. Find the simplified form of $\sqrt{8} \cdot \sqrt{50}$.
3. Rationalize the denominator of $\frac{7}{\sqrt{2} + 1}$.
4. Simplify $\sqrt{24 \cdot 6}$.
5. Expand and simplify $(\sqrt{5} + 2)^2$.

Tip:

Always look for perfect squares when simplifying square roots to make calculations easier!