To simplify the expression [tex]\(\sqrt{7} \cdot \sqrt{2}\)[/tex], we can make use of the property of square roots that states:
[tex]\[ \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \][/tex]
Here, we need to identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex]. In our case, [tex]\(a = 7\)[/tex] and [tex]\(b = 2\)[/tex]. According to the property mentioned above, we can combine these under a single square root:
[tex]\[ \sqrt{7} \cdot \sqrt{2} = \sqrt{7 \cdot 2} \][/tex]
Next, we calculate the product inside the square root:
[tex]\[ 7 \cdot 2 = 14 \][/tex]
So the expression simplifies to:
[tex]\[ \sqrt{14} \][/tex]
To verify the accuracy of our simplification and to understand the numerical values, we can approximate the value of [tex]\(\sqrt{14}\)[/tex]. Using a calculator or referencing known values, we find:
[tex]\[ \sqrt{14} \approx 3.7416573867739413 \][/tex]
Thus, the simplified form of [tex]\(\sqrt{7} \cdot \sqrt{2}\)[/tex] is [tex]\(\sqrt{14}\)[/tex], and its approximate value is:
[tex]\[ \sqrt{14} \approx 3.7416573867739413 \][/tex]
