Simplify: $(-7-\sqrt{7})(-3+\sqrt{7})$
To simplify the expression \((-7 - \sqrt{7})(-3 + \sqrt{7})\), we use the distributive property of multiplication over addition, commonly known as the FOIL method (First, Outer, Inner, Last). The expression is a product of two binomials of the form \((a + b)(c + d)\), which expands to \(ac + ad + bc + bd\). Here, let \(a = -7\), \(b = -\sqrt{7}\), \(c = -3\), and \(d = \sqrt{7}\).
We apply the distributive property:
\((-7 - \sqrt{7})(-3 + \sqrt{7}) = (-7)(-3) + (-7)(\sqrt{7}) + (-\sqrt{7})(-3) + (-\sqrt{7})(\sqrt{7})\)
Now, we perform the individual multiplications:
Summing these results gives:
\(21 - 7\sqrt{7} + 3\sqrt{7} - 7\)
Now, we combine the rational terms and the terms involving \(\sqrt{7}\):
Combining these, we get:
\(14 - 4\sqrt{7}\)
Comparing this result to the given options, we see that it matches the third option.
In engineering and physics, calculating the product of radical expressions often arises when dealing with impedance in AC circuits or calculating geometric properties of objects containing symmetry. For instance, if you are calculating the side lengths or areas of a structure involving diagonal components in a grid, you may encounter expressions similar to this one, where simplifying helps determine the final physical dimensions or values needed for design.