To determine the product [tex]\((\sqrt{14} - \sqrt{3})(\sqrt{12} + \sqrt{7})\)[/tex], we use the distributive property (also known as the FOIL method for binomials). The distributive property states that:
[tex]\[
(a + b)(c + d) = ac + ad + bc + bd
\][/tex]
Applying this to our expression:
[tex]\[
(\sqrt{14} - \sqrt{3})(\sqrt{12} + \sqrt{7}) = \sqrt{14}\sqrt{12} + \sqrt{14}\sqrt{7} - \sqrt{3}\sqrt{12} - \sqrt{3}\sqrt{7}
\][/tex]
Let's break down each term one by one:
1. [tex]\(\sqrt{14} \cdot \sqrt{12}\)[/tex]
2. [tex]\(\sqrt{14} \cdot \sqrt{7}\)[/tex]
3. [tex]\(-\sqrt{3} \cdot \sqrt{12}\)[/tex]
4. [tex]\(-\sqrt{3} \cdot \sqrt{7}\)[/tex]
First term:
[tex]\[
\sqrt{14} \cdot \sqrt{12} = \sqrt{14 \cdot 12} = \sqrt{168} \approx 12.96148139681572
\][/tex]
Second term:
[tex]\[
\sqrt{14} \cdot \sqrt{7} = \sqrt{14 \cdot 7} = \sqrt{98} \approx 9.899494936611665
\][/tex]
Third term:
[tex]\[
-\sqrt{3} \cdot \sqrt{12} = -\sqrt{3 \cdot 12} = -\sqrt{36} = -6
\][/tex]
Fourth term:
[tex]\[
-\sqrt{3} \cdot \sqrt{7} = -\sqrt{3 \cdot 7} = -\sqrt{21} \approx -4.58257569495584
\][/tex]
Now, we can combine these terms to get the final expression:
[tex]\[
(\sqrt{14} - \sqrt{3})(\sqrt{12} + \sqrt{7}) = \sqrt{168} + \sqrt{98} - 6 - \sqrt{21}
\][/tex]
[tex]\[
\approx 12.96148139681572 + 9.899494936611665 - 6 - 4.58257569495584
\][/tex]
Therefore, we simplify and recognize the similar terms:
[tex]\[
12.96148139681572 - 4.58257569495584 = 2\sqrt{42} - \sqrt{21}
\][/tex]
Hence, the simplified form of the expression [tex]\((\sqrt{14} - \sqrt{3})(\sqrt{12} + \sqrt{7})\)[/tex] is:
[tex]\[
2\sqrt{42} - \sqrt{21}
\][/tex]
