Practice section 7 of 8

Simplify $\dfrac{y^{2/3}-y^{-2/3}}{y^{-4/3}-1}$.

Solve the equation $9^{2x+1}-27^x=0$.

Solve the equation $\log_x(2x+3)=2$.

Solve the equation $3^{x+1}\cdot4^{x+2}=5^{x+3}$.

Express each of the following in partial fractions.

(a) $\dfrac4{x^2-4}$

(b) $\dfrac{x^2-11}{(x+2)^2(3x-1)}$

(c) $\dfrac{x^4+1}{x^3+2x}$

Given that $\theta=\sin^{-1}(x/2)$, where $x>0$, find in terms of $x$ (i) $\cos\theta$ (ii) $\sin2\theta$.