Exercise Set 1

These exercises cover the topics of Algebra.

  1. Simplify each of the following (Hint: use the rules of exponents where needed).

    1. \(x = 3pq+5pr=2qr+qp-6rp\)
    2. \(y = 5l^2mn+2nl^2m-3mln^2+l^2nm+4n^2ml-nm^2\)
    3. \(z = \dfrac{(s^\frac{1}{3})^\frac{3}{4}\times (t^\frac{1}{4})^{-1}}{(t^\frac{1}{2}\times (s^{-\frac{1}{4}})^{-1})}\)

  2. Expand the brackets in each of the following and simplify the expression.

    1. \(-4x(2x-y)(3x+2y)\)
    2. \((a-2b)(2a-3b)(3a-4b)\)
    3. \(-\{-2[x-3(y-4)]-5(z+6)\}\)
    4. \((v^3-v^2-2)(1-3v+2v^2)\)

  3. Simplify each of the following.

    1. \(\dfrac{p}{q^3}\div\dfrac{p^3}{q}\)
    2. \(\dfrac{a^2b}{2c}\times\dfrac{ac^2}{2b}\div\dfrac{b^2c}{2a}\)
    3. \(\dfrac{8x^{-3}\times 3x^2}{6x^{-4}}\)
    4. \(\dfrac{3x}{3x^2+6x}\)

  4. Factorise the following expressions.

    1. \(18x^2y-12xy^2\)
    2. \(x^3+4x^2y-3xy^2-12y^3\)
    3. \(25x^2-4y^2\)

  5. The characteristic equation of a perfect gas is given by \(P V = mRT\) where \(m\) is the mass, \(P\) is the pressure, \(V\) is the volume, \(T\) is the temperature and \(R\) is the universal gas constant. Make temperature the subject of the formula.

  6. The airflow over a turbine blade causes drag \(D\), which is given by \(D= \dfrac{\rho C v^2 A}{2}\), where \(\rho\) is fluid density, \(C\) is the drag coefficient, \(v\) is fluid velocity and \(A\) is the frontal area of the blade. Make the frontal area the subject of the formula.

  7. Make \(b\) the subject of the following formula. \[W=\frac{t\sqrt{a+b^2}}{2\pi}\]