Question

(-2square root 77 + 8square root 21)(-6square root 77 - 3square root 14)

Answer 1

To simplify the expression \((-2\sqrt{77} + 8\sqrt{21})(-6\sqrt{77} - 3\sqrt{14})\), we'll use the distributive property (also known as the FOIL method for binomials).

Let:
- \( a = -2\sqrt{77} \)
- \( b = 8\sqrt{21} \)
- \( c = -6\sqrt{77} \)
- \( d = -3\sqrt{14} \)

Now, we can multiply the terms:

\[
(a + b)(c + d) = ac + ad + bc + bd
\]

Calculating each term:

1. **Calculate \( ac \)**:
\[
ac = (-2\sqrt{77})(-6\sqrt{77}) = 12 \cdot 77 = 924
\]

2. **Calculate \( ad \)**:
\[
ad = (-2\sqrt{77})(-3\sqrt{14}) = 6\sqrt{77 \cdot 14} = 6\sqrt{1078}
\]

3. **Calculate \( bc \)**:
\[
bc = (8\sqrt{21})(-6\sqrt{77}) = -48\sqrt{1617}
\]

4. **Calculate \( bd \)**:
\[
bd = (8\sqrt{21})(-3\sqrt{14}) = -24\sqrt{294}
\]

Now, combine all the terms together:
\[
924 + 6\sqrt{1078} - 48\sqrt{1617} - 24\sqrt{294}
\]

Thus, the final expression is:
\[
924 + 6\sqrt{1078} - 48\sqrt{1617} - 24\sqrt{294}
\]

This is the simplified form of the original expression.