Let's solve the problem. The expression is:

$3\sqrt{2} \times \sqrt{5}$

We need to simplify this expression and rewrite it in the form $\sqrt{t}$, where $t$ is an integer.

Step 1: Combine the square roots

Using the property of square roots, $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$, we have:

$\sqrt{2} \cdot \sqrt{5} = \sqrt{2 \cdot 5} = \sqrt{10}.$

Step 2: Multiply by the coefficient

Now, multiply $3$ by $\sqrt{10}$:

$3 \sqrt{10}.$

This is already in the desired form, where $t = 10$.

Final Answer:

$t = 10$

Would you like further clarification or a breakdown of specific steps?

Related Questions:

1. What are the rules for multiplying and simplifying square roots?
2. Can all square roots be simplified further? If not, why?
3. What happens if there are coefficients outside the square root in both terms?
4. How can we generalize this method for other problems involving square roots?
5. How is this type of problem applied in real-world situations like physics or engineering?

Tip:

When working with square roots, always look for opportunities to simplify by factoring perfect squares out of the radicand.