What is the following product?

[tex]\[ (\sqrt{14} - \sqrt{3})(\sqrt{12} + \sqrt{7}) \][/tex]

A. \( 2\sqrt{42} + 7\sqrt{2} - 6 - \sqrt{21} \)
B. \( \sqrt{14} - 6 + \sqrt{7} \)
C. \( \sqrt{26} + \sqrt{21} - \sqrt{15} - \sqrt{10} \)
D. [tex]\( 2\sqrt{42} - \sqrt{21} \)[/tex]



Answer :

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]