Question

(-3square root 26 + 4square root 35)(- square root 65 + 4square root 14)

Answer 1

To simplify the expression \((-3\sqrt{26} + 4\sqrt{35})(-\sqrt{65} + 4\sqrt{14})\), we will use the distributive property (also known as the FOIL method for binomials).

Let's denote:

\[
A = -3\sqrt{26} + 4\sqrt{35}
\]
\[
B = -\sqrt{65} + 4\sqrt{14}
\]

Now, we need to multiply \(A\) and \(B\):

\[
AB = (-3\sqrt{26})(-\sqrt{65}) + (-3\sqrt{26})(4\sqrt{14}) + (4\sqrt{35})(-\sqrt{65}) + (4\sqrt{35})(4\sqrt{14})
\]

Calculating each term:

1. **First Term:**
\[
(-3\sqrt{26})(-\sqrt{65}) = 3\sqrt{26 \cdot 65} = 3\sqrt{1690}
\]

2. **Second Term:**
\[
(-3\sqrt{26})(4\sqrt{14}) = -12\sqrt{26 \cdot 14} = -12\sqrt{364} = -12\cdot 2\sqrt{91} = -24\sqrt{91}
\]

3. **Third Term:**
\[
(4\sqrt{35})(-\sqrt{65}) = -4\sqrt{35 \cdot 65} = -4\sqrt{2275}
\]

4. **Fourth Term:**
\[
(4\sqrt{35})(4\sqrt{14}) = 16\sqrt{35 \cdot 14} = 16\sqrt{490} = 16\cdot 7\sqrt{10} = 112\sqrt{10}
\]

Putting it all together:

\[
AB = 3\sqrt{1690} - 24\sqrt{91} - 4\sqrt{2275} + 112\sqrt{10}
\]

Thus, the fully simplified expression is:

\[
3\sqrt{1690} - 24\sqrt{91} - 4\sqrt{2275} + 112\sqrt{10}
\]

This is the simplified form of the given expression.