To simplify the expression [tex]\( 3 \sqrt{50} + 2 \sqrt{45} - \sqrt{2} + \sqrt{5} \)[/tex], follow these steps:
1. Simplify the square roots individually:
- [tex]\( \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5 \sqrt{2} \)[/tex]
- [tex]\( \sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3 \sqrt{5} \)[/tex]
- [tex]\( \sqrt{2} \)[/tex] remains [tex]\( \sqrt{2} \)[/tex]
- [tex]\( \sqrt{5} \)[/tex] remains [tex]\( \sqrt{5} \)[/tex]
2. Substitute these simplified square roots back into the expression:
- [tex]\( 3 \sqrt{50} = 3 \times 5 \sqrt{2} = 15 \sqrt{2} \)[/tex]
- [tex]\( 2 \sqrt{45} = 2 \times 3 \sqrt{5} = 6 \sqrt{5} \)[/tex]
Thus, the expression now becomes:
[tex]\[
15 \sqrt{2} + 6 \sqrt{5} - \sqrt{2} + \sqrt{5}
\][/tex]
3. Combine like terms:
- Combine the [tex]\(\sqrt{2}\)[/tex] terms: [tex]\( 15 \sqrt{2} - \sqrt{2} = (15 - 1) \sqrt{2} = 14 \sqrt{2} \)[/tex]
- Combine the [tex]\(\sqrt{5}\)[/tex] terms: [tex]\( 6 \sqrt{5} + \sqrt{5} = (6 + 1) \sqrt{5} = 7 \sqrt{5} \)[/tex]
4. Write the final simplified expression:
The simplified expression is:
[tex]\[
7 \sqrt{5} + 14 \sqrt{2}
\][/tex]
Hence, [tex]\( 3 \sqrt{50} + 2 \sqrt{45} - \sqrt{2} + \sqrt{5} \)[/tex] simplifies to [tex]\( 7 \sqrt{5} + 14 \sqrt{2} \)[/tex].
