Simplify the following surds :1) 2√21 × √27 ÷ √3432) 7√5 × √125 ÷ 2√27

Answer:1) [tex]\frac{18}{7}[/tex]2) [tex]\frac{175\sqrt{3}}{18}[/tex]Step-by-step explanation:* Lets explain how to simplify a square root1)∵ [tex]2\sqrt{21}[/tex] × [tex]\sqrt{27}[/tex] ÷ [tex]\sqrt{343}[/tex]∵ [tex]\sqrt{21}=\sqrt{3}[/tex] × [tex]\sqrt{7}[/tex]∴ [tex]2\sqrt{21}[/tex] = [tex]2\sqrt{3}[/tex] × [tex]\sqrt{7}[/tex]∵ [tex]\sqrt{27}[/tex] = [tex]\sqrt{3}[/tex] × [tex]\sqrt{3}[/tex] × [tex]\sqrt{3}[/tex]∵ [tex]\sqrt{3}[/tex] × [tex]\sqrt{3}[/tex] = 3∴ [tex]\sqrt{27}[/tex] = [tex]3\sqrt{3}[/tex]∴ [tex]2\sqrt{21}[/tex] × [tex]\sqrt{27}[/tex] =   [tex]2\sqrt{3}[/tex] × [tex]\sqrt{7}[/tex] × [tex]3\sqrt{3}[/tex]∵ [tex]\sqrt{3}[/tex] × [tex]\sqrt{3}[/tex] = 3∵ 2 × 3 × 3 = 18∴ [tex]2\sqrt{21}[/tex] × [tex]\sqrt{27}[/tex] = [tex]18\sqrt{7}[/tex]∵ [tex]\sqrt{343}[/tex] = [tex]\sqrt{7}[/tex] × [tex]\sqrt{7}[/tex] × [tex]\sqrt{7}[/tex]∵ [tex]\sqrt{7}[/tex] × [tex]\sqrt{7}[/tex] = 7∴ [tex]\sqrt{343}[/tex] = [tex]7\sqrt{7}[/tex]∵ [tex]2\sqrt{21}[/tex] × [tex]\sqrt{27}[/tex] ÷ [tex]\sqrt{343}[/tex] =   [tex]18\sqrt{7}[/tex] ÷ [tex]7\sqrt{7}[/tex]∵ [tex]\sqrt{7}[/tex] ÷ [tex]\sqrt{7}[/tex] = 1∴ [tex]2\sqrt{21}[/tex] × [tex]\sqrt{27}[/tex] ÷ [tex]\sqrt{343}[/tex] =   [tex]\frac{18}{7}[/tex] 2) ∵ [tex]7\sqrt{5}[/tex] × [tex]\sqrt{125}[/tex] ÷ [tex]2\sqrt{27}[/tex]   ∵ [tex]\sqrt{125}[/tex] = [tex]\sqrt{5}[/tex] × [tex]\sqrt{5}[/tex] × [tex]\sqrt{5}[/tex] ∵ [tex]\sqrt{5}[/tex] × [tex]\sqrt{5}[/tex] = 5∴ [tex]\sqrt{125}[/tex] = [tex]5\sqrt{5}[/tex]∴ [tex]7\sqrt{5}[/tex] × [tex]\sqrt{125}[/tex] =   [tex]7\sqrt{5}[/tex] × [tex]5\sqrt{5}[/tex]∵ [tex]\sqrt{5}[/tex] × [tex]\sqrt{5}[/tex] = 5∴ [tex]7\sqrt{5}[/tex] × [tex]\sqrt{125}[/tex] = 7 × 5 × 5 = 175∵ [tex]2\sqrt{27}[/tex] = [tex]2\sqrt{3}[/tex] × [tex]\sqrt{3}[/tex] × [tex]\sqrt{3}[/tex]∵ [tex]\sqrt{3}[/tex] × [tex]\sqrt{3}[/tex] = 3∴ [tex]2\sqrt{27}[/tex] = [tex]6\sqrt{3}[/tex]∴ [tex]7\sqrt{5}[/tex] × [tex]\sqrt{125}[/tex] ÷ [tex]2\sqrt{27}[/tex] =   175 ÷ [tex]6\sqrt{3}[/tex] = [tex]\frac{175}{6\sqrt{3}}[/tex]∵ [tex]\frac{175}{6\sqrt{3}}[/tex] not in the simplest form because   the denominator has square root∴ Multiply up and down by [tex]\sqrt{3}[/tex]∴  [tex]\frac{175}{6\sqrt{3}}[/tex] = [tex]\frac{175\sqrt{3}}{6\sqrt{3}*\sqrt{3}}[/tex]∴  [tex]\frac{175}{6\sqrt{3}}[/tex] = [tex]\frac{175\sqrt{3}}{18}[/tex]∴ [tex]7\sqrt{5}[/tex] × [tex]\sqrt{125}[/tex] ÷ [tex]2\sqrt{27}[/tex] =   [tex]\frac{175\sqrt{3}}{18}[/tex]