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Let's solve the expression \( 3\sqrt{12} \times \sqrt{5} \), given in the simplest form as \( a\sqrt{b} \), where \( a \) and \( b \) are integers.

### Step 1: Simplify \( \sqrt{12} \)
We can express 12 as a product of its prime factors:
\[
\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}
\]

### Step 2: Substitute into the original expression
Now we substitute \( \sqrt{12} \) into the given expression:
\[
3\sqrt{12} \times \sqrt{5} = 3 \times (2\sqrt{3}) \times \sqrt{5} = 6\sqrt{3} \times \sqrt{5}
\]

### Step 3: Combine the square roots
Using the property \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \), we get:
\[
6\sqrt{3} \times \sqrt{5} = 6\sqrt{15}
\]

Thus, the expression simplifies to \( 6\sqrt{15} \).

### Conclusion:
The expression \( 3\sqrt{12} \times \sqrt{5} \) in its simplest form is \( 6\sqrt{15} \). Therefore, the values of \( a \) and \( b \) are:
- \( a = 6 \)
- \( b = 15 \)

Would you like more details or have any questions about this solution?

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Here are 5 related questions to expand on this problem:
1. How do you simplify other expressions with square roots, like \( \sqrt{50} \times \sqrt{2} \)?
2. What are the properties of square roots that make simplification possible?
3. Can we simplify \( \sqrt{72} \times \sqrt{18} \) in a similar way?
4. What happens if you have cube roots instead of square roots?
5. How do these simplifications apply to real-world problems involving radicals?

**Tip:** When simplifying square roots, always look for perfect square factors to make the process easier.

The answer to 3√12 × √5, given in its simplest form, is a√b, where a and b are integers. Calculate the values of a and b.

Discover how to simplify the expression 3√12 × √5 into its simplest form a√b. This problem covers simplifying square roots and multiplying radicals. Perfect for students in Grades 8-10.

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