To simplify the product [tex]\((3 + \sqrt{7})(3 - \sqrt{7})\)[/tex], we'll use the difference of squares formula, which states:
[tex]\[
(a + b)(a - b) = a^2 - b^2
\][/tex]
In this case, let [tex]\( a = 3 \)[/tex] and [tex]\( b = \sqrt{7} \)[/tex]. Applying the formula:
[tex]\[
(3 + \sqrt{7})(3 - \sqrt{7}) = 3^2 - (\sqrt{7})^2
\][/tex]
First, calculate [tex]\( 3^2 \)[/tex]:
[tex]\[
3^2 = 9
\][/tex]
Next, calculate [tex]\((\sqrt{7})^2\)[/tex]:
[tex]\[
(\sqrt{7})^2 = 7
\][/tex]
Now, substitute these values back into the formula:
[tex]\[
(3 + \sqrt{7})(3 - \sqrt{7}) = 9 - 7
\][/tex]
Finally, perform the subtraction:
[tex]\[
9 - 7 = 2
\][/tex]
Thus, the simplified product [tex]\((3 + \sqrt{7})(3 - \sqrt{7})\)[/tex] is:
[tex]\[
\boxed{2}
\][/tex]
