If 3\sqrt{3}3 = 1.732, then find the value of :
(i)27+75+108−243(ii)512−348+675+7108\begin{matrix} \text{(i)} & \sqrt{27} + \sqrt{75} + \sqrt{108} - \sqrt{243} \\[1.5em] \text{(ii)} & 5\sqrt{12} - 3\sqrt{48} + 6\sqrt{75} + 7\sqrt{108} \\[1.5em] \end{matrix}(i)(ii)27+75+108−243512−348+675+7108
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(i) 27+75+108−243=3×3×3+3×5×5+2×2×3×3×3−3×3×3×3×3=33+53+63−93=(3+5+6−9)×3=(14−9)×3=5×1.732=8.660\text{(i) } \sqrt{27} + \sqrt{75} + \sqrt{108} - \sqrt{243} \\[1.5em] = \sqrt{3 × 3 × 3 } + \sqrt{3 × 5 × 5} + \sqrt{2 × 2 × 3 × 3 ×3 } - \sqrt{3 × 3 × 3 × 3 × 3} \\[1.5em] = 3\sqrt{3} + 5\sqrt{3} + 6\sqrt{3} - 9\sqrt{3} \\[1.5em] = (3 + 5 + 6 - 9) × \sqrt{3} \\[1.5em] = (14 - 9) × \sqrt{3} \\[1.5em] = \bold{5 × 1.732 = 8.660} \\[1.5em](i) 27+75+108−243=3×3×3+3×5×5+2×2×3×3×3−3×3×3×3×3=33+53+63−93=(3+5+6−9)×3=(14−9)×3=5×1.732=8.660
(ii) 512−348+675+7108=52×2×3−32×2×2×2×3+65×5×3+72×2×3×3×3=5×23−4×33+6×53+7×63=103−123+303+423=(10−12+30+42)×3=(82−12)×3=70×1.732=121.24\text{(ii) } 5\sqrt{12} - 3\sqrt{48} + 6\sqrt{75} + 7\sqrt{108} \\[1.5em] = 5\sqrt{2 × 2 × 3 } - 3\sqrt{2 × 2 × 2 × 2 × 3} + 6\sqrt{5 × 5 × 3 } + 7\sqrt{2 × 2 × 3 × 3 × 3} \\[1.5em] = 5 × 2\sqrt{3} - 4 × 3\sqrt{3} + 6 × 5\sqrt{3} + 7 × 6\sqrt{3} \\[1.5em] = 10\sqrt{3} - 12\sqrt{3} + 30\sqrt{3} + 42\sqrt{3} \\[1.5em] = (10 - 12 + 30 +42) × \sqrt{3} \\[1.5em] = (82 - 12) × \sqrt{3} \\[1.5em] = \bold{70 × 1.732 = 121.24} \\[1.5em](ii) 512−348+675+7108=52×2×3−32×2×2×2×3+65×5×3+72×2×3×3×3=5×23−4×33+6×53+7×63=103−123+303+423=(10−12+30+42)×3=(82−12)×3=70×1.732=121.24
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Simplify the following:
(i)(5+7)(2+5)(ii)(5+5)(5−5)(iii)(5+2)2(iv)(3−7)2(v)(2+3)(5+7)(vi)(4+5)(3−7)\begin{matrix} \text{(i)} & (5 + \sqrt{7})(2 + \sqrt{5}) \\[1.5em] \text{(ii)} & (5 + \sqrt{5})(5 - \sqrt{5}) \\[1.5em] \text{(iii)} & (\sqrt{5} + \sqrt{2})^2 \\[1.5em] \text{(iv)} & (\sqrt{3} - \sqrt{7})^2 \\[1.5em] \text{(v)} & (\sqrt{2} + \sqrt{3})(\sqrt{5} + \sqrt{7}) \\[1.5em] \text{(vi)} & (4 + \sqrt{5})(\sqrt{3} - \sqrt{7}) \\[1.5em] \end{matrix}(i)(ii)(iii)(iv)(v)(vi)(5+7)(2+5)(5+5)(5−5)(5+2)2(3−7)2(2+3)(5+7)(4+5)(3−7)
If 2\sqrt{2}2=1.414, then find the value of :
(i)8+50+72+98(ii)332−250+4128−2018\begin{matrix} \text{(i)} & \sqrt{8} + \sqrt{50} + \sqrt{72} + \sqrt{98} \\[1.5em] \text{(ii)} & 3\sqrt{32} - 2\sqrt{50} + 4\sqrt{128} - 20\sqrt{18} \\[1.5em] \end{matrix}(i)(ii)8+50+72+98332−250+4128−2018
State which of the following numbers are irrational :
(i)49,−370,725,165(ii)−249,3200,253,−4916\begin{matrix} \text{(i)} & \sqrt{\dfrac{4}{9}}, -{\dfrac{3}{70}},\sqrt{\dfrac{7}{25}},\sqrt{\dfrac{16}{5}} \\[1.5em] \text{(ii)} & -{\sqrt{\dfrac{2}{49}}}, {\dfrac{3}{200}},\sqrt{\dfrac{25}{3}},-{\sqrt{\dfrac{49}{16}}} \\[1.5em] \end{matrix}(i)(ii)94,−703,257,516−492,2003,325,−1649
State which of the following numbers will change into non-terminating, non-recurring decimals :
(i)−32(ii)25681(iii)27×16(iv)536\begin{matrix} \text{(i)} & - 3\sqrt{2} \\[1.5em] \text{(ii)} & \sqrt{\dfrac{256}{81}} \\[1.5em] \text{(iii)} & \sqrt{27 × 16} \\[1.5em] \text{(iv)} & \sqrt{\dfrac{5}{36}} \\[1.5em] \end{matrix}(i)(ii)(iii)(iv)−328125627×16365
