Simplify : (3sqrt{75} – sqrt{12} + sqrt{108}), leaving the answer in surd form (radicals)….

Explanation

(a) (3sqrt{75} = 3 times sqrt{25 times 3} = 3(5sqrt{3}) = 15sqrt{3})

(sqrt{12} = sqrt{4 times 3} = 2sqrt{3})

(sqrt{108} = sqrt{36 times 3} = 6sqrt{3})

(therefore 3sqrt{75} – sqrt{12} + sqrt{108} equiv 15sqrt{3} – 2sqrt{3} + 6sqrt{3})

= ((15 – 2 + 6)sqrt{3})

= (19sqrt{3}).

(b) (124_{n} = 232_{five})

(124_{n} = (1 times n^{2}) + (2 times n^{1}) + (4 times n^{0}) )

= (n^{2} + 2n + 4)

(232_{five} = (2 times 5^{2}) + (3 times 5^{1}) + (2 times 5^{0}))

= (50 + 15 + 2)

= (67)

(implies n^{2} + 2n + 4 = 67)

(n^{2} + 2n + 4 – 67 = 0)

(n^{2} + 2n – 63 = 0)

(n^{2} – 7n + 9n – 63 = 0)

(n(n – 7) + 9(n – 7) = 0)

((n + 9)(n – 7) = 0)

(text{n = -9 or 7})

Hence, (n = 7)

(124_{seven} = 232_{five})