Question

Multiply (7–√) (−2–√). Give an exact answer.(1 point)
Responses

−72–√
negative 7 start root 2 end root

−14−−√
negative start root 14 end root

5–√
start root 5 end root

14−−√
start root 14 end root

Answer 1

To multiply \((7 - \sqrt{2})(-2 - \sqrt{2})\), we can use the distributive property (also known as the FOIL method for binomials). Let's perform the multiplication step by step:

1. First, multiply the first terms:
\[
7 \cdot (-2) = -14
\]

2. Next, multiply the outer terms:
\[
7 \cdot (-\sqrt{2}) = -7\sqrt{2}
\]

3. Now, multiply the inner terms:
\[
-\sqrt{2} \cdot (-2) = 2\sqrt{2}
\]

4. Finally, multiply the last terms:
\[
-\sqrt{2} \cdot (-\sqrt{2}) = 2
\]

Now, combine all these results together:
\[
-14 - 7\sqrt{2} + 2 + 2\sqrt{2}
\]

Next, combine like terms:
- The constant terms: \(-14 + 2 = -12\)
- The terms with \(\sqrt{2}\): \(-7\sqrt{2} + 2\sqrt{2} = -5\sqrt{2}\)

Thus, the final result is:
\[
-12 - 5\sqrt{2}
\]

None of the options provided exactly matches this answer, but if we were to express it in a simplified format, it would be \(-12 - 5\sqrt{2}\). If you see an option that closely resembles this (particularly in terms of square roots combined with constants), that would be the correct response.

However, based on the given options:
- None of them are a perfect match to \(-12 - 5\sqrt{2}\).

If we were to round up the operation, the closest representation for this would not be accurately covered in the options provided.