Certainly! Let's tackle part (i) of your question in detail:
To simplify the expression: [tex]\(\sqrt{45}-3 \sqrt{20}+4 \sqrt{5}\)[/tex]
Step-by-step solution:
1. Simplify each square root term individually:
- [tex]\(\sqrt{45}\)[/tex]:
[tex]\[
\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}
\][/tex]
- [tex]\(-3 \sqrt{20}\)[/tex]:
[tex]\[
-3 \sqrt{20} = -3 \times \sqrt{4 \times 5} = -3 \times \sqrt{4} \times \sqrt{5} = -3 \times 2 \sqrt{5} = -6 \sqrt{5}
\][/tex]
- [tex]\(4 \sqrt{5}\)[/tex] is already in its simplest form.
2. Substitute the simplified terms back into the original expression:
[tex]\[
\sqrt{45} - 3 \sqrt{20} + 4 \sqrt{5} = 3 \sqrt{5} - 6 \sqrt{5} + 4 \sqrt{5}
\][/tex]
3. Combine like terms (terms involving [tex]\(\sqrt{5}\)[/tex]):
[tex]\[
3 \sqrt{5} - 6 \sqrt{5} + 4 \sqrt{5} = (3 - 6 + 4) \sqrt{5} = 1 \sqrt{5} = \sqrt{5}
\][/tex]
So, the simplified form of the expression is:
[tex]\[
\sqrt{5}
\][/tex]
Final Answer:
[tex]\[
\sqrt{45} - 3 \sqrt{20} + 4 \sqrt{5} = \sqrt{5}
\][/tex]
Feel free to reach out if you have any questions on part (ii) or any other queries!
