WAEC 2023 · Paper 2 · Q3

(a)
Express $\dfrac{7 + 2\sqrt{10}}{9\sqrt5 + 5\sqrt2}$ in the form $p\sqrt5 + q\sqrt2$ .
Show the answer
$\frac{43}{355}\sqrt5 + \frac{11}{71}\sqrt2$
(b)
Using the values of $p$ and $q$ in 3(a), find the value of $(2p - q)$ .

Worked solution (try it first)

Multiply the top and the bottom by the conjugate of the bottom, $9\sqrt5 - 5\sqrt2$ .
The bottom: $(9\sqrt5)^2 - (5\sqrt2)^2 = 405 - 50$
$= 355$ .
The top: $(7 + 2\sqrt{10})(9\sqrt5 - 5\sqrt2) = 63\sqrt5 - 35\sqrt2 + 18\sqrt{50} - 10\sqrt{20}$ .
Simplify the surds: $18\sqrt{50} = 90\sqrt2$ and $10\sqrt{20} = 20\sqrt5$ .
So the top is $43\sqrt5 + 55\sqrt2$ .
Divide each term by 355: $p = \dfrac{43}{355}$ and $q = \dfrac{55}{355} = \dfrac{11}{71}$ .