To find the value of [tex]\((\sqrt{14} - \sqrt{3})(\sqrt{12} + \sqrt{7})\)[/tex], let's proceed by expanding the product using the distributive property, often referred to as the FOIL method (First, Outer, Inner, Last).
[tex]\[
(\sqrt{14} - \sqrt{3})(\sqrt{12} + \sqrt{7}) = \sqrt{14} \cdot \sqrt{12} + \sqrt{14} \cdot \sqrt{7} - \sqrt{3} \cdot \sqrt{12} - \sqrt{3} \cdot \sqrt{7}
\][/tex]
We will handle each term separately:
1. [tex]\(\sqrt{14} \cdot \sqrt{12} = \sqrt{14 \cdot 12} = \sqrt{168}\)[/tex]
2. [tex]\(\sqrt{14} \cdot \sqrt{7} = \sqrt{14 \cdot 7} = \sqrt{98}\)[/tex]
3. [tex]\(\sqrt{3} \cdot \sqrt{12} = \sqrt{3 \cdot 12} = \sqrt{36} = 6\)[/tex]
4. [tex]\(\sqrt{3} \cdot \sqrt{7} = \sqrt{3 \cdot 7} = \sqrt{21}\)[/tex]
So, combining all terms, we get:
[tex]\[
\sqrt{168} + \sqrt{98} - 6 - \sqrt{21}
\][/tex]
Now, substituting the approximate numerical values for each term:
- [tex]\(\sqrt{168} \approx 12.961\)[/tex]
- [tex]\(\sqrt{98} \approx 9.899\)[/tex]
- [tex]\(\sqrt{36} = 6\)[/tex]
- [tex]\(\sqrt{21} \approx 4.583\)[/tex]
Combining these approximations:
[tex]\[
12.961 + 9.899 - 6 - 4.583
\][/tex]
Swatch out fractions:
[tex]\[
12.961 + 9.899 - 4.583 - 6 = 12.961 + 5.316
\][/tex]
Thus, we simplify:
[tex]\[
18.277
\][/tex]
Now, comparing this value with the given choices:
- [tex]\(2 \sqrt{42} + 7 \sqrt{2} - 6 - \sqrt{21}\)[/tex]
- [tex]\(\sqrt{14} - 6 + \sqrt{7}\)[/tex]
- [tex]\(\sqrt{26} + \sqrt{21} - \sqrt{15} - \sqrt{10}\)[/tex]
- [tex]\(2 \sqrt{42} - \sqrt{21}\)[/tex]
Upon examination, none of these options exactly match the simplified product value. Therefore, it appears the accurate expression contributing closely aligns as the detailed numbers concurs are fulfilled by matching applied respective approximations towards matching scenarios distinctly.
