### Part (a):
1. Simplify [tex]\(\sqrt{12}\)[/tex] in terms of [tex]\(\sqrt{3}\)[/tex]:
[tex]\[\sqrt{12} = \sqrt{4 \times 3} \][/tex]
[tex]\[\sqrt{12} = \sqrt{4} \times \sqrt{3} \][/tex]
[tex]\[\sqrt{12} = 2 \sqrt{3} \][/tex]
2. Add [tex]\(\sqrt{3}\)[/tex] and [tex]\(\sqrt{12}\)[/tex] in terms of [tex]\(\sqrt{3}\)[/tex]:
[tex]\[\sqrt{3} + \sqrt{12} = \sqrt{3} + 2 \sqrt{3} \][/tex]
[tex]\[\sqrt{3} + \sqrt{12} = (1 + 2) \sqrt{3} \][/tex]
[tex]\[\sqrt{3} + \sqrt{12} = 3 \sqrt{3} \][/tex]
So, the answer to part (a) is [tex]\(3 \sqrt{3}\)[/tex].
### Part (b)(i):
1. Simplify [tex]\(\left(\frac{1}{\sqrt{3}}\right)^7\)[/tex]:
[tex]\[\left(\frac{1}{\sqrt{3}}\right)^7 = \frac{1^7}{(\sqrt{3})^7} \][/tex]
[tex]\[\left(\frac{1}{\sqrt{3}}\right)^7 = \frac{1}{(\sqrt{3})^7} \][/tex]
2. Express [tex]\(\frac{1}{(\sqrt{3})^7}\)[/tex] in the form [tex]\(\frac{1}{b \sqrt{3}}\)[/tex]:
[tex]\[\sqrt{3}^7 = (\sqrt{3})^6 \times \sqrt{3} \][/tex]
[tex]\[\sqrt{3}^7 = (3^{3}) \times \sqrt{3} \][/tex]
[tex]\[\sqrt{3}^7 = 27 \sqrt{3} \][/tex]
Therefore,
[tex]\[\frac{1}{\sqrt{3}^7} = \frac{1}{27 \sqrt{3}} \][/tex]
[tex]\[\frac{1}{b \sqrt{3}} \text{ where } b = 27\][/tex]
So, the answer to part (b)(i) is [tex]\(\frac{1}{27 \sqrt{3}}\)[/tex], where [tex]\(b = 27\)[/tex].
### Part (b)(ii):
1. Rewrite [tex]\(\left(\frac{1}{\sqrt{3}}\right)^7\)[/tex] in a simpler form:
[tex]\[\left(\frac{1}{\sqrt{3}}\right)^7 = \frac{1}{(\sqrt{3})^7} \][/tex]
As established before,
[tex]\[\left(\frac{1}{\sqrt{3}}\right)^7 = \frac{1}{27 \sqrt{3}} \][/tex]
2. Express [tex]\(\frac{1}{27 \sqrt{3}}\)[/tex] in the form [tex]\(\frac{\sqrt{3}}{c}\)[/tex]:
[tex]\[\frac{1}{27 \sqrt{3}} = \frac{\sqrt{3}}{27 \times 3}\][/tex]
[tex]\[\frac{1}{27 \sqrt{3}} = \frac{\sqrt{3}}{81}\][/tex]
So, the answer to part (b)(ii) is [tex]\(\frac{\sqrt{3}}{81}\)[/tex], where [tex]\(c = 81\)[/tex].
