Question

(8 + 6square root 5)(2 - 2square root 10)

Answer 1

To simplify the expression \((8 + 6\sqrt{5})(2 - 2\sqrt{10})\), we can use the distributive property (also known as the FOIL method for binomials). We'll distribute each term in the first binomial by each term in the second binomial.

Let's calculate it step-by-step:

1. **Multiply the first terms:**
\[
8 \times 2 = 16
\]

2. **Multiply the outer terms:**
\[
8 \times (-2\sqrt{10}) = -16\sqrt{10}
\]

3. **Multiply the inner terms:**
\[
6\sqrt{5} \times 2 = 12\sqrt{5}
\]

4. **Multiply the last terms:**
\[
6\sqrt{5} \times (-2\sqrt{10}) = -12\sqrt{50}
\]

Since \(\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}\), we have:
\[
-12\sqrt{50} = -12 \times 5\sqrt{2} = -60\sqrt{2}
\]

Now we can combine all these results together:

\[
16 - 16\sqrt{10} + 12\sqrt{5} - 60\sqrt{2}
\]

Therefore, the simplified result is:

\[
\boxed{16 + 12\sqrt{5} - 16\sqrt{10} - 60\sqrt{2}}
\]