$($a$)$ Using de Moivre's Theorem, express the following complex number in the form $a+bi$ :

$(\sqrt[\phantom{2}]{5}+4i{)}^6$

b$)$ Evaluate the following limits:

i$)$

$\underset{x}{lim}\frac{7x^3-2x^7-2x^4+1}{2x^3-7x^7+12x+2x^5}$

ii$)$

$\underset{x9}{lim}\frac{x^2-81}{x-9}$

c$)$ Differentiate the following function from first principles:

$f(x)=6x^2-3x+7$

d$)$ Simplify the following expressions:

i$)$

$\frac{\sqrt[4]{a^7}(a^5)}{a^{\frac{4}{3}}}$

ii$)$

$\frac{(3-x{)}^3(-x^2-2x+15)}{(x^2-6x+9{)}^2}$

e$)$ Find the equation of the line $L$ through the points $(2,-4)$ and $(-2,-2).$ Find also the equation of the line $K$ which is perpendicular to $L$ and passes through the point $(-1,2).$

f$)$ Given $f(x)=(2x^2+1{)}^3$ and $g(x)=\frac{1}{4x},$ find:

$(f$@$g)(x),(g$@$f)(x)$