Answer :

To simplify the expression [tex]\(\sqrt{6} \cdot \sqrt{12}\)[/tex], we can follow these steps:

1. Express the product of square roots as a single square root:

[tex]\[ \sqrt{6} \cdot \sqrt{12} = \sqrt{6 \times 12} \][/tex]

2. Calculate the product inside the square root:

[tex]\[ 6 \times 12 = 72 \][/tex]

So the expression becomes:

[tex]\[ \sqrt{72} \][/tex]

3. Simplify the square root:

Notice that [tex]\(72\)[/tex] can be factored into [tex]\(36 \times 2\)[/tex], and [tex]\(36\)[/tex] is a perfect square. Hence, we can write:

[tex]\[ \sqrt{72} = \sqrt{36 \times 2} \][/tex]

Using the property of square roots that [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex], we get:

[tex]\[ \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} \][/tex]

4. Calculate the square root of the perfect square:

[tex]\(\sqrt{36} = 6\)[/tex], so:

[tex]\[ \sqrt{36} \times \sqrt{2} = 6 \times \sqrt{2} \][/tex]

Thus, the simplified form of [tex]\(\sqrt{6} \cdot \sqrt{12}\)[/tex] is:

[tex]\[ 6 \sqrt{2} \][/tex]

5. Verify the numerical values for accuracy:

To ensure correctness, we can use numerical approximations for the original expression and the simplified form.

- Approximating [tex]\(\sqrt{6} \approx 2.449489742783178\)[/tex]
- Approximating [tex]\(\sqrt{12} \approx 3.4641016151377544\)[/tex]
- Multiplying these: [tex]\(2.449489742783178 \times 3.4641016151377544 \approx 8.48528137423857\)[/tex]

The simplified form is:

- [tex]\(6 \times \sqrt{2} \approx 6 \times 1.4142135623730951 = 8.48528137423857\)[/tex]

Both methods yield the same numerical result, confirming that the simplification is correct.

Therefore, the simplified form of [tex]\(\sqrt{6} \cdot \sqrt{12}\)[/tex] is:

[tex]\[ 6\sqrt{2} \][/tex]