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Let's solve the problem step-by-step by simplifying each surd expression given:

### (i) \(\sqrt{6} \times \sqrt{27}\)

To simplify this, we use the property of square roots: \(\sqrt{a} \times \sqrt{b} = \sqrt{a \cdot b}\).

\[
\sqrt{6} \times \sqrt{27} = \sqrt{6 \cdot 27}
\]

Now, compute the product:

\[
6 \cdot 27 = 162, \text{ so we have } \sqrt{162}.
\]

Now, simplify \(\sqrt{162}\) by factoring it into its prime factors:

\[
162 = 2 \times 3^4.
\]

\[
\sqrt{162} = \sqrt{2 \times 3^4} = \sqrt{2} \times 3^2 = 9\sqrt{2}.
\]

So, \(\sqrt{6} \times \sqrt{27} = 9\sqrt{2}\).

### (ii) \(\sqrt{12} \times \sqrt{15}\)

Again, use the property of square roots:

\[
\sqrt{12} \times \sqrt{15} = \sqrt{12 \cdot 15}.
\]

Compute the product:

\[
12 \cdot 15 = 180, \text{ so we have } \sqrt{180}.
\]

Now, simplify \(\sqrt{180}\) by factoring it into its prime factors:

\[
180 = 2^2 \times 3^2 \times 5.
\]

\[
\sqrt{180} = \sqrt{2^2 \times 3^2 \times 5} = 2 \times 3 \times \sqrt{5} = 6\sqrt{5}.
\]

So, \(\sqrt{12} \times \sqrt{15} = 6\sqrt{5}\).

### (iii) \(\sqrt{10} \times \sqrt{24} \times \sqrt{15}\)

Combine the square roots:

\[
\sqrt{10} \times \sqrt{24} \times \sqrt{15} = \sqrt{10 \cdot 24 \cdot 15}.
\]

Compute the product:

\[
10 \cdot 24 \cdot 15 = 3600, \text{ so we have } \sqrt{3600}.
\]

Now, simplify \(\sqrt{3600}\):

\[
3600 = 60^2, \text{ so } \sqrt{3600} = 60.
\]

So, \(\sqrt{10} \times \sqrt{24} \times \sqrt{15} = 60\).

### Final Simplified Results:
1. \(\sqrt{6} \times \sqrt{27} = 9\sqrt{2}\)
2. \(\sqrt{12} \times \sqrt{15} = 6\sqrt{5}\)
3. \(\sqrt{10} \times \sqrt{24} \times \sqrt{15} = 60\)

Would you like more details on any of these steps or have any questions?

Here are 5 related questions to expand your understanding:
1. How can you simplify expressions that involve cube roots?
2. What are some other properties of square roots that can help with simplification?
3. How do you handle square roots with fractions?
4. Can you find an alternative method to simplify \(\sqrt{180}\)?
5. What is the difference between surds and irrational numbers in general?

**Tip:** Always look for common factors or perfect squares within the radicand when simplifying square roots.

Write these in terms of the simplest possible surd: (i) √6 × √27, (ii) √12 × √15, (iii) √10 × √24 × √15.

Learn how to simplify surds such as √6 × √27, √12 × √15, and √10 × √24 × √15 using properties of square roots and prime factorization. This guide covers essential concepts in radical simplification, perfect square factors, and multiplication of radicals, suitable for high school students.

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