Question

Multiply. Write your answer in simplest form.
(4 squareroot35 + 8 squareroot22) (-3squareroot14 - 9 squareroot10)

Answer 1

To multiply the expression \((4\sqrt{35} + 8\sqrt{22})(-3\sqrt{14} - 9\sqrt{10})\), we will use the distributive property (also known as the FOIL method for binomials).

Let's distribute each term in the first expression to each term in the second expression.

1. **Multiply \(4\sqrt{35}\) by \(-3\sqrt{14}\)**:
\[
4\sqrt{35} \cdot (-3\sqrt{14}) = -12\sqrt{35 \cdot 14} = -12\sqrt{490}
\]

2. **Multiply \(4\sqrt{35}\) by \(-9\sqrt{10}\)**:
\[
4\sqrt{35} \cdot (-9\sqrt{10}) = -36\sqrt{35 \cdot 10} = -36\sqrt{350}
\]

3. **Multiply \(8\sqrt{22}\) by \(-3\sqrt{14}\)**:
\[
8\sqrt{22} \cdot (-3\sqrt{14}) = -24\sqrt{22 \cdot 14} = -24\sqrt{308}
\]

4. **Multiply \(8\sqrt{22}\) by \(-9\sqrt{10}\)**:
\[
8\sqrt{22} \cdot (-9\sqrt{10}) = -72\sqrt{22 \cdot 10} = -72\sqrt{220}
\]

Now, we combine all these results:
\[
-12\sqrt{490} - 36\sqrt{350} - 24\sqrt{308} - 72\sqrt{220}
\]

Next, we simplify the square roots where possible.

1. **Simplifying \(\sqrt{490}\)**:
\[
\sqrt{490} = \sqrt{49 \cdot 10} = 7\sqrt{10}
\]
Thus,
\[
-12\sqrt{490} = -12 \cdot 7\sqrt{10} = -84\sqrt{10}
\]

2. **Simplifying \(\sqrt{350}\)**:
\[
\sqrt{350} = \sqrt{25 \cdot 14} = 5\sqrt{14}
\]
Thus,
\[
-36\sqrt{350} = -36 \cdot 5\sqrt{14} = -180\sqrt{14}
\]

3. **Simplifying \(\sqrt{308}\)**:
\[
\sqrt{308} = \sqrt{4 \cdot 77} = 2\sqrt{77}
\]
Thus,
\[
-24\sqrt{308} = -24 \cdot 2\sqrt{77} = -48\sqrt{77}
\]

4. **Simplifying \(\sqrt{220}\)**:
\[
\sqrt{220} = \sqrt{4 \cdot 55} = 2\sqrt{55}
\]
Thus,
\[
-72\sqrt{220} = -72 \cdot 2\sqrt{55} = -144\sqrt{55}
\]

Now we can combine all the results into a single expression:
\[
-84\sqrt{10} - 180\sqrt{14} - 48\sqrt{77} - 144\sqrt{55}
\]

Thus, the final answer in its simplest form is:
\[
\boxed{-84\sqrt{10} - 180\sqrt{14} - 48\sqrt{77} - 144\sqrt{55}}
\]