Question
(8square root 10 + square root 55)(square root 55 + 10square root 22)
Answers
GPT-4o mini
To simplify the expression \( (8\sqrt{10} + \sqrt{55})(\sqrt{55} + 10\sqrt{22}) \), we can use the distributive property (also known as the FOIL method for binomials).
1. **Distribute \(8\sqrt{10}\)**:
- \(8\sqrt{10} \cdot \sqrt{55} = 8 \sqrt{550}\)
- \(8\sqrt{10} \cdot 10\sqrt{22} = 80\sqrt{220}\)
2. **Distribute \(\sqrt{55}\)**:
- \(\sqrt{55} \cdot 8\sqrt{10} = 8\sqrt{550}\) (this term is the same as above)
- \(\sqrt{55} \cdot 10\sqrt{22} = 10\sqrt{1210}\)
3. **Now we combine everything**:
So the expression expands to:
\[
8\sqrt{550} + 80\sqrt{220} + 10\sqrt{1210}
\]
Since we have \(8\sqrt{550}\) from both terms of \(8\sqrt{10}\) and \(\sqrt{55}\), we can add those two:
\[
(8\sqrt{550} + 8\sqrt{550}) + 80\sqrt{220} + 10\sqrt{1210} = 16\sqrt{550} + 80\sqrt{220} + 10\sqrt{1210}
\]
4. **We have reached the final result!** The simplified expression is:
\[
\boxed{16\sqrt{550} + 80\sqrt{220} + 10\sqrt{1210}}
\]
1. **Distribute \(8\sqrt{10}\)**:
- \(8\sqrt{10} \cdot \sqrt{55} = 8 \sqrt{550}\)
- \(8\sqrt{10} \cdot 10\sqrt{22} = 80\sqrt{220}\)
2. **Distribute \(\sqrt{55}\)**:
- \(\sqrt{55} \cdot 8\sqrt{10} = 8\sqrt{550}\) (this term is the same as above)
- \(\sqrt{55} \cdot 10\sqrt{22} = 10\sqrt{1210}\)
3. **Now we combine everything**:
So the expression expands to:
\[
8\sqrt{550} + 80\sqrt{220} + 10\sqrt{1210}
\]
Since we have \(8\sqrt{550}\) from both terms of \(8\sqrt{10}\) and \(\sqrt{55}\), we can add those two:
\[
(8\sqrt{550} + 8\sqrt{550}) + 80\sqrt{220} + 10\sqrt{1210} = 16\sqrt{550} + 80\sqrt{220} + 10\sqrt{1210}
\]
4. **We have reached the final result!** The simplified expression is:
\[
\boxed{16\sqrt{550} + 80\sqrt{220} + 10\sqrt{1210}}
\]