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]
