To simplify the expression [tex]\(\sqrt{3} \cdot \sqrt{7}\)[/tex], we first need to understand the properties of square roots and their operations. Specifically, we can use the property that the product of square roots can be expressed as the square root of the product of the radicands:
[tex]\[
\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}
\][/tex]
Applying this property to our expression, we get:
[tex]\[
\sqrt{3} \cdot \sqrt{7} = \sqrt{3 \cdot 7}
\][/tex]
Now, we perform the multiplication inside the square root:
[tex]\[
\sqrt{3 \cdot 7} = \sqrt{21}
\][/tex]
Next, we determine the approximate value of [tex]\(\sqrt{21}\)[/tex]. The precise steps to calculate this value involve knowing the numerical approximations for [tex]\(\sqrt{3}\)[/tex] and [tex]\(\sqrt{7}\)[/tex], and then multiplying them together:
[tex]\[
\sqrt{3} \approx 1.7320508075688772
\][/tex]
[tex]\[
\sqrt{7} \approx 2.6457513110645907
\][/tex]
Multiplying these two values together, we get:
[tex]\[
1.7320508075688772 \cdot 2.6457513110645907 \approx 4.58257569495584
\][/tex]
Therefore, the simplified form of [tex]\(\sqrt{3} \cdot \sqrt{7}\)[/tex] is:
[tex]\[
\sqrt{21} \approx 4.58257569495584
\][/tex]
Thus, the numerical approximation for the product [tex]\(\sqrt{3} \cdot \sqrt{7}\)[/tex] is approximately [tex]\(4.58257569495584\)[/tex].
