The problem asks to simplify the expression $\frac{\sqrt{60}}{\sqrt{5}}$. Here's how to solve it step by step:

Step 1: Apply the quotient rule for square roots

The quotient rule states that $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$. Applying this rule, we have: $\frac{\sqrt{60}}{\sqrt{5}} = \sqrt{\frac{60}{5}}$

Step 2: Simplify the fraction under the square root

$\frac{60}{5} = 12$ So the expression becomes: $\sqrt{12}$

Step 3: Simplify $\sqrt{12}$

Break 12 into its prime factors: $12 = 4 \cdot 3$ Since 4 is a perfect square, we can simplify: $\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}$

Final Answer:

$\frac{\sqrt{60}}{\sqrt{5}} = 2\sqrt{3}$

Would you like more details or explanations? Here are 5 related questions to expand your understanding:

1. How can you simplify an expression like $\frac{\sqrt{a}}{\sqrt{b}}$ for other numbers?
2. Why does the quotient rule for square roots work mathematically?
3. Can you prove that $\sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3}$?
4. How do you handle expressions involving cube roots or higher-order roots?
5. What are some other ways to simplify square root expressions without using prime factorization?

Tip:

When simplifying square roots, always check for perfect squares as factors—they make simplification much easier!