Factorise and simplify the following algebraic fractions:
\dfrac{6 x - 16}{12}
\dfrac{x - 4}{4 - x}
\dfrac{6 \left(p - q\right)^{2}}{6 p^{2} - 6 q^{2}}
\dfrac{2 x^{2} + 10 x - 100}{2 x + 20}
\dfrac{2 x + 10 y}{x + 5 y}
\dfrac{x-1}{x^{2} - 2 x + 1}
\dfrac{2 r - 8}{r^{2} - 16}
\dfrac{x^{2} + 14 x + 49}{x^{2} + 4 x - 21}
\dfrac{4 x^{2} + 7 x - 2}{6 + x - x^{2}}
\dfrac{x^{2} + x y + x z + y z}{x^{2} + 2 x y + y^{2}}
\dfrac{5 \left(k^{2} - 5\right)^{4} + 20 k \left(k^{2} - 5\right)^{5}}{15 \left(k^{2} - 5\right)^{4}}
Complete the following:
Find the lowest common denominator of \dfrac{19}{4 m^{2}} and \dfrac{5}{12 m}.
Hence, write \dfrac{19}{4 m^{2}} + \dfrac{5}{12 m} as a single simplified fraction.
Simplify:
\dfrac{x}{5} + \dfrac{4 x}{7}
\dfrac{4}{7 p} - \dfrac{2}{p}
- 3 x - \dfrac{3 x}{8}
\dfrac{7 x}{11} + \dfrac{3 x - 2}{44}
\dfrac{4 x}{7} - \dfrac{2 x - 3}{21}
\dfrac{5 x - y}{x - y} + \dfrac{x - 6 y}{y - x}
\dfrac{5 x}{7 \left(x + 1\right)} - \dfrac{10}{x + 1}
\dfrac{1}{5} + \dfrac{7}{x - 6}
\dfrac{2 y + 5}{y - 3} - \dfrac{y + 3}{y - 3}
Consider the pair of fractions:
\dfrac{17}{6 t^{2} + 30 t + 36} \quad \text{ and } \quad \dfrac{14 + 3 t}{t^{2} + 5 t + 6}
Factorise t^{2} + 5 t + 6
Factorise 6 t^{2} + 30 t + 36
Hence, find the lowest common denominator in factorised form.
Simplify:
\dfrac{2}{\left(x + 3\right) \left(x - 10\right)} + \dfrac{5}{\left(x + 3\right) \left(x + 5\right)}
\dfrac{x}{x^{2} - 16} - \dfrac{12}{x + 4}
\dfrac{x^{2} - 5}{x^{2} - 4 x - 32} - 1 - \dfrac{2}{x - 8}
\dfrac{x^{2} + 11 x + 18}{x^{2} - 4} - \dfrac{x + 5}{x^{2} + 3 x - 10}
\dfrac{4}{m^{2} - 7 m - 18} - \dfrac{3}{m^{2} - 4 m - 45} + \dfrac{5}{m^{2} - 81}
\dfrac{\left(y - 1\right) \left(y + 5\right)}{\left(y + 1\right) \left(y - 5\right)} - \dfrac{\left(y + 3\right) \left(y + 4\right)}{\left(y + 1\right) \left(5 - y\right)} - \dfrac{\left(y + 2\right) \left(y - 6\right)}{\left(y + 1\right) \left(5 - y\right)}
Simplify:
\dfrac{5 u}{6 a} \times \dfrac{3 v}{20 b}
\dfrac{5 y}{2} \times 6 y^{5}
\dfrac{x^{2}}{5} \times \dfrac{15}{x}
\dfrac{3 y^{2}}{4} \times \dfrac{12}{29 y^{6}}
\dfrac{27 - 27 x}{28} \times \dfrac{4}{9 x - 9}
\dfrac{5 x + 8}{8 x y^{2}} \times \dfrac{9 x y}{25 x + 40}
\dfrac{p + 7}{5} \times \dfrac{5 p - 2}{p^{2} + 14 p + 49}
\dfrac{9 x y + 18 x}{x - 9} \times \dfrac{x y - 3 x - 9 y + 27}{4 y^{2} + 8 y}
The product of two fractions is 1. If one of the fractions is \dfrac{5 x}{9 y}, what is the other fraction?
Find the value of K in the expression:
K \div \dfrac{5 m}{9} = \dfrac{3 n}{m}
Simplify:
\dfrac{m}{4} \div \dfrac{13}{10}
\dfrac{3 y^{2}}{4} \div \dfrac{12}{29 y^{6}}
\dfrac{- 9 n}{4} \div \dfrac{11 n}{8}
\dfrac{2 u}{6 v} \div \dfrac{7 v}{12 u}
\dfrac{16 u}{17 v} \div \left( - 14 u v \right)
\left(\dfrac{1}{x} + \dfrac{1}{y}\right) \div \left(x + y\right)
Factorise and simplify the following expressions:
\dfrac{k - 2}{5} \div \dfrac{4 k - 8}{15}
\dfrac{k - 1}{2} \div \dfrac{k^{2} - 1}{6}
\dfrac{3 m^{2} + 3 m}{7 m y z} \div \dfrac{10 m + 10}{3 m z}
\dfrac{\left(m + 5\right) \left(m + 6\right)}{25 \left(m + 8\right)} \div \dfrac{\left(m + 6\right) \left(m + 8\right)}{10 \left(m + 8\right)}
\dfrac{p^{2} + 5 q + p q + 5 p}{9 q p^{2} + 45 q} \times \dfrac{9 p^{2} + 45}{p^{2} + 14 p + 45}
\dfrac{5 x^{2} + 22 x + 8}{x^{2} - 16} \div \dfrac{25 x^{2} - 4}{20 x^{2} - 8 x}
\dfrac{6 p + 48}{p^{2} - 64} \div \dfrac{12 + 12 p}{p^{2} - 2 p - 48}
What expression is missing from the equation below?
\dfrac{4 k^{2} - 13 k - 12}{k^{2} - 2 k - 8} \times ⬚ \times \dfrac{1}{\left( 4 k + 3\right)} = 1
Simplify the following expressions:
\dfrac{3 x y}{12 y} \times \dfrac{4 y^{2}}{3 w x} \div \dfrac{15 w x}{3 w^{2}}
\left(\dfrac{q}{5} + \dfrac{q}{10}\right) \times \left(\dfrac{q}{5} - \dfrac{q}{10}\right)
\dfrac{x + 3}{x + 4} \div \dfrac{x + 3}{3} - \dfrac{1}{x + 4}
\left(\dfrac{2}{x + 1} - \dfrac{4}{\left(x + 1\right)^{2}}\right) \div \dfrac{x + 5}{\left(x + 1\right)^{2}}
A rectangle has an area of \dfrac{5 x^{3} y^{4}}{3 p q} square units and a length of \dfrac{4 x y}{p} units. Find an expression for the width of the rectangle.
Simplify:
\dfrac{\frac{4}{t} + 9}{\frac{4}{t} - 9}
\dfrac{a^{ - 1 } + 4}{a^{ - 1 } - 4}
\dfrac{\frac{7}{x - 1} + 1}{1 - \frac{6}{x - 1}}
\dfrac{\frac{3}{x} - \frac{1}{x y}}{\frac{1}{x y} + \frac{7}{y}}
Simplify the expression \dfrac{\frac{1}{3} - \frac{1}{x}}{\frac{7}{6} + \frac{1}{x^{2}}}. Give your answer in factorised form.