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
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(i) 8+50+72+98=2×2×2+5×5×2+6×6×2+2×7×7=22+52+62+72=(2+5+6+7)×2=(20)×2=20×1.414=28.28\text{(i) } \sqrt{8} + \sqrt{50} + \sqrt{72} + \sqrt{98} \\[1.5em] = \sqrt{2 × 2 × 2} + \sqrt{5 × 5 × 2} + \sqrt{6 × 6 × 2} + \sqrt{2 × 7 × 7} \\[1.5em] = 2\sqrt{2} + 5\sqrt{2} + 6\sqrt{2} + 7\sqrt{2} \\[1.5em] = (2 + 5 + 6 + 7) × \sqrt{2} \\[1.5em] = (20) × \sqrt{2} \\[1.5em] = \bold{20 × 1.414 = 28.28 } \\[1.5em](i) 8+50+72+98=2×2×2+5×5×2+6×6×2+2×7×7=22+52+62+72=(2+5+6+7)×2=(20)×2=20×1.414=28.28
(ii) 332−250+4128−2018=32×4×4−25×5×2+48×8×2−202×3×3=122−102+322−602=(12−10+32−60)×2=(44−70)×2=(−26)×2=−26×1.414=−36.764\text{(ii) } 3\sqrt{32} - 2\sqrt{50} + 4\sqrt{128} - 20\sqrt{18} \\[1.5em] = 3\sqrt{2 × 4 × 4 } - 2\sqrt{5 × 5 × 2} + 4\sqrt{8 × 8 × 2} - 20\sqrt{2 × 3 × 3} \\[1.5em] = 12\sqrt{2} - 10\sqrt{2} + 32\sqrt{2} - 60\sqrt{2} \\[1.5em] = (12 - 10 + 32 - 60) × \sqrt{2} \\[1.5em] = (44 - 70) × \sqrt{2} \\[1.5em] = (-26) × \sqrt{2} \\[1.5em] = \bold{-26 × 1.414 = -36.764 } \\[1.5em](ii) 332−250+4128−2018=32×4×4−25×5×2+48×8×2−202×3×3=122−102+322−602=(12−10+32−60)×2=(44−70)×2=(−26)×2=−26×1.414=−36.764
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Simplify the following:
(i)45−320+45(ii)33+227+73(iii)65×25(iv)815÷23(v)248+549(vi)38+12\begin{matrix} \text{(i)} & \sqrt{45} - 3\sqrt{20} + 4\sqrt{5} \\[1.5em] \text{(ii)} & 3\sqrt{3} + 2\sqrt{27} + \dfrac{7}{\sqrt{3}} \\[1.5em] \text{(iii)} & 6\sqrt{5} × 2\sqrt{5} \\[1.5em] \text{(iv)} & 8\sqrt{15} ÷ 2\sqrt{3} \\[1.5em] \text{(v)} & \dfrac{\sqrt{24}}{8} + \dfrac{\sqrt{54}}{9} \\[1.5em] \text{(vi)} & \dfrac{3}{\sqrt{8}} + \dfrac{1}{\sqrt{2}} \\[1.5em] \end{matrix}(i)(ii)(iii)(iv)(v)(vi)45−320+4533+227+3765×25815÷23824+95483+21
(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 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
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
