2a) Rationalize the following surds then solve.

The rationalized expression is $\boxed{\frac{5\sqrt{3}}{3}}$.

ii) $\frac{3}{\sqrt{3}}$

The rationalized expression is $\boxed{\sqrt{3}}$.

iii) $\frac{4}{\sqrt{2}}$

The rationalized expression is $\boxed{2\sqrt{2}}$.

iv) $\frac{1}{\sqrt{2}-\sqrt{5}}$

Remember $(a-b)(a+b) = a^2-b^2$.

The rationalized expression is $\boxed{-\frac{\sqrt{2}+\sqrt{5}}{3}}$.

2b) Simplify the following Surds.

i) $\sqrt{32} + 9\sqrt{8}$

ii) $2\sqrt{20} + \sqrt{125} - 3\sqrt{5}$

The simplified expression is $\boxed{6\sqrt{5}}$.

iii) $\sqrt{98} - \sqrt{32} - 2\sqrt{2}$

The simplified expression is $\boxed{\sqrt{2}}$.

2c) Integrate all with respect to x the following simple equation.

i) $8x^2$

The integral is $\boxed{\frac{8}{3}x^3 + C}$.

ii) $\int \frac{dx}{3x+2}$

Here $a=3$ and $b=2$.

The integral is $\boxed{\frac{1}{3}\ln|3x+2| + C}$.

iii) $\frac{x^4+x+4}{x^3}$

The integral is $\boxed{\frac{x^2}{2} - \frac{1}{x} - \frac{2}{x^2} + C}$.

2d) Write down The Conjugate of the Surds.

i) $\sqrt{3}+\sqrt{2}$

The conjugate of $\sqrt{3}+\sqrt{2}$ is $\boxed{\sqrt{3}-\sqrt{2}}$.

ii) $\frac{13+5\sqrt{2}}{7}$

The conjugate of the surd part $13+5\sqrt{2}$ is $\boxed{13-5\sqrt{2}}$. (The denominator does not change for the conjugate of the expression itself, only the sign of the surd term in the numerator).

iii) $1-2+\sqrt{5}$

The conjugate of $-1+\sqrt{5}$ is $\boxed{-1-\sqrt{5}}$.

iv) $5\sqrt{7}+5\sqrt{3}+\sqrt{7}-\sqrt{3}$

The conjugate of $6\sqrt{7}+4\sqrt{3}$ is $\boxed{6\sqrt{7}-4\sqrt{3}}$.