Sure! Let's break down and solve each part of the given problem step-by-step.
### Part (a) Simplify the following surds
#### i. [tex]\( 2 \sqrt{18} + 3 \sqrt{48} - \sqrt{50} - 2 \sqrt{27} \)[/tex]
1. Simplify each term individually:
- [tex]\( 2 \sqrt{18} \)[/tex]
We can rewrite [tex]\(\sqrt{18}\)[/tex] as [tex]\(\sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}\)[/tex].
Hence, [tex]\(2 \sqrt{18} = 2 \times 3 \sqrt{2} = 6 \sqrt{2}\)[/tex].
- [tex]\( 3 \sqrt{48} \)[/tex]
We can rewrite [tex]\(\sqrt{48}\)[/tex] as [tex]\(\sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3}\)[/tex].
Hence, [tex]\(3 \sqrt{48} = 3 \times 4 \sqrt{3} = 12 \sqrt{3}\)[/tex].
- [tex]\( - \sqrt{50} \)[/tex]
We can rewrite [tex]\(\sqrt{50}\)[/tex] as [tex]\(\sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}\)[/tex].
Hence, [tex]\( - \sqrt{50} = -5 \sqrt{2} \)[/tex].
- [tex]\( -2 \sqrt{27} \)[/tex]
We can rewrite [tex]\(\sqrt{27}\)[/tex] as [tex]\(\sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3}\)[/tex].
Hence, [tex]\(-2 \sqrt{27} = -2 \times 3 \sqrt{3} = -6 \sqrt{3}\)[/tex].
2. Combine the simplified terms:
Putting it all together:
[tex]\[
6 \sqrt{2} + 12 \sqrt{3} - 5 \sqrt{2} - 6 \sqrt{3}
\][/tex]
Group the like terms:
[tex]\[
(6 \sqrt{2} - 5 \sqrt{2}) + (12 \sqrt{3} - 6 \sqrt{3}) = \sqrt{2} + 6 \sqrt{3}
\][/tex]
Hence, the simplified form is:
[tex]\[
\sqrt{2} + 6 \sqrt{3}
\][/tex]
#### ii. [tex]\(\frac{\sqrt{2} + \sqrt{5}}{\sqrt{10}}\)[/tex]
1. Rationalize the denominator:
Multiply the numerator and denominator by [tex]\(\sqrt{10}\)[/tex] to eliminate the square root in the denominator.
[tex]\[
\frac{\sqrt{2} + \sqrt{5}}{\sqrt{10}} \times \frac{\sqrt{10}}{\sqrt{10}} = \frac{(\sqrt{2} + \sqrt{5}) \sqrt{10}}{10}
\][/tex]
2. Simplify the numerator:
Distribute [tex]\(\sqrt{10}\)[/tex] to each term inside the parentheses in the numerator:
[tex]\[
(\sqrt{2} \times \sqrt{10}) + (\sqrt{5} \times \sqrt{10}) = \sqrt{20} + \sqrt{50}
\][/tex]
Now, rewrite [tex]\(\sqrt{20}\)[/tex] and [tex]\(\sqrt{50}\)[/tex]:
[tex]\[
\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5},\quad \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}
\][/tex]
Hence,
[tex]\[
\sqrt{20} + \sqrt{50} = 2\sqrt{5} + 5\sqrt{2}
\][/tex]
Therefore,
[tex]\[
\frac{2\sqrt{5} + 5\sqrt{2}}{10}
\][/tex]
3. Combine and simplify if necessary:
Divide each term by 10:
[tex]\[
\frac{2\sqrt{5}}{10} + \frac{5\sqrt{2}}{10} = \frac{\sqrt{5}}{5} + \frac{\sqrt{2}}{2}
\][/tex]
Hence, the simplified form is approximately:
[tex]\[
1.1543203766865056
\][/tex]
This concludes the solution for part ii.
### Summary:
1. [tex]\( 2 \sqrt{18} + 3 \sqrt{48} - \sqrt{50} - 2 \sqrt{27} = \sqrt{2} + 6 \sqrt{3} \approx 11.80651840778636\)[/tex]
2. [tex]\(\frac{\sqrt{2} + \sqrt{5}}{\sqrt{10}} \approx 1.1543203766865056\)[/tex]
Keep these steps in mind when you tackle similar problems involving surds!
