Manipulating Expressions — practice questions

Expand and simplify $(3x - 4)^2$.

Factorise completely $4x^3 - 36x$.

A rectangle has a length that is 3 cm more than twice its width. A square of side $x$ cm is cut from each corner and the sides folded up to make an open box. Given the width of the rectangle is $w$ cm, write an expression in factorised form for the volume of the box, and state the restrictions on $x$.

Simplify $\dfrac{x^2 - 16}{x^2 + 8x + 16}$.

Express $\dfrac{4}{x-2} - \dfrac{3}{x+1}$ as a single fraction in its simplest form.

Show that $\dfrac{1}{x} - \dfrac{1}{x+h}$ simplifies to $\dfrac{h}{x(x+h)}$, and hence find an expression for $\dfrac{1}{h}\left(\dfrac{1}{x} - \dfrac{1}{x+h}\right)$ in its simplest form. State what happens to this expression as $h$ becomes very small.

Evaluate $16^{3/4}$ and $5^{-2}$, giving exact answers.

Simplify $\dfrac{\sqrt{x^5} \times x^{-1/2}}{x^{2}}$, giving your answer as a single power of $x$.

Solve $x^{2/3} - 5x^{1/3} + 6 = 0$ for $x$, giving all solutions exactly.