To find the value of the expression \((\sqrt{14} - \sqrt{3})(\sqrt{12} + \sqrt{7})\), let's break down the steps in a detailed manner.
First, note that:
- \(\sqrt{14} \approx 3.741657386773941\)
- \(\sqrt{3} \approx 1.7320508075688772\)
- \(\sqrt{12} \approx 3.4641016151377544\)
- \(\sqrt{7} \approx 2.6457513110645907\)
Now we can evaluate each component step-by-step:
1. Calculate \(\sqrt{14} - \sqrt{3}\):
[tex]\[
\sqrt{14} - \sqrt{3} \approx 3.741657386773941 - 1.7320508075688772 = 2.009606579205064
\][/tex]
2. Calculate \(\sqrt{12} + \sqrt{7}\):
[tex]\[
\sqrt{12} + \sqrt{7} \approx 3.4641016151377544 + 2.6457513110645907 = 6.109852926202345
\][/tex]
Next, we need to find the product of these two expressions:
[tex]\[
(\sqrt{14} - \sqrt{3})(\sqrt{12} + \sqrt{7}) = 2.009606579205064 \times 6.109852926202345
\][/tex]
When we multiply these numbers, we get:
[tex]\[
2.009606579205064 \times 6.109852926202345 = 12.278400638471545
\][/tex]
Therefore, the product \((\sqrt{14} - \sqrt{3})(\sqrt{12} + \sqrt{7})\) evaluates to approximately \(12.278400638471545\).
Thus, the value of the given expression is:
[tex]\[
\boxed{12.278400638471545}
\][/tex]
