Question Number 139278 by Agnibhoo last updated on 25/Apr/21
$$\sqrt{\frac{\mathrm{3000000}\:+\:\mathrm{3000000}}{\mathrm{4000000}\:+\:\mathrm{4000000}}}\:=\:? \\ $$
Answered by Rasheed.Sindhi last updated on 25/Apr/21
$$\sqrt{\frac{\mathrm{6}×\cancel{\mathrm{1000000}}}{\mathrm{8}×\cancel{\mathrm{1000000}}}}=\frac{\sqrt{\mathrm{3}}}{\mathrm{2}} \\ $$
Commented by otchereabdullai@gmail.com last updated on 25/Apr/21
$$\mathrm{nice}\:\mathrm{shortcut} \\ $$
Answered by Opredador last updated on 25/Apr/21
$$\: \\ $$$$\:\sqrt{\frac{\mathrm{6000000}}{\mathrm{8000000}}}\:\Rightarrow\:\sqrt{\frac{\mathrm{6}×\mathrm{10}^{\mathrm{6}} }{\mathrm{8}×\mathrm{10}^{\mathrm{6}} }}\:\Rightarrow\:\frac{\sqrt{\mathrm{6}×\mathrm{10}^{\mathrm{6}} }}{\:\sqrt{\mathrm{8}×\mathrm{10}^{\mathrm{6}} }} \\ $$$$\:\Rightarrow\:\frac{\sqrt{\mathrm{6}}×\sqrt{\mathrm{10}^{\mathrm{6}} }}{\:\sqrt{\mathrm{8}}×\sqrt{\mathrm{10}^{\mathrm{6}} }}\:\Rightarrow\:\frac{\sqrt{\mathrm{6}}×\mathrm{10}^{\frac{\mathrm{6}}{\mathrm{2}}} }{\:\sqrt{\mathrm{8}}×\mathrm{10}^{\frac{\mathrm{6}}{\mathrm{2}}} }\:\Rightarrow\:\frac{\sqrt{\mathrm{6}}×\mathrm{10}^{\mathrm{3}} }{\:\sqrt{\mathrm{8}}×\mathrm{10}^{\mathrm{3}} } \\ $$$$\:\Rightarrow\:\frac{\mathrm{10}^{\mathrm{3}} }{\mathrm{10}^{\mathrm{3}} }×\frac{\sqrt{\mathrm{6}}}{\:\sqrt{\mathrm{8}}}×\frac{\sqrt{\mathrm{8}}}{\:\sqrt{\mathrm{8}}}\:\Rightarrow\:\frac{\sqrt{\mathrm{6}}×\sqrt{\mathrm{8}}}{\:\sqrt{\mathrm{8}×\mathrm{8}}}\:\Rightarrow\:\frac{\sqrt{\mathrm{6}}×\sqrt{\mathrm{2}^{\mathrm{3}} }}{\:\sqrt{\mathrm{8}^{\mathrm{2}} }} \\ $$$$\:\Rightarrow\:\frac{\sqrt{\mathrm{6}}×\sqrt{\mathrm{2}^{\mathrm{2}} ×\mathrm{2}}}{\mathrm{8}}\:\Rightarrow\:\frac{\sqrt{\mathrm{6}}×\sqrt{\mathrm{2}^{\mathrm{2}} }×\sqrt{\mathrm{2}}}{\mathrm{8}}\:\Rightarrow\:\frac{\sqrt{\mathrm{6}}×\mathrm{2}×\sqrt{\mathrm{2}}}{\mathrm{2}×\mathrm{4}} \\ $$$$\:\Rightarrow\:\frac{\sqrt{\mathrm{6}}×\cancel{\mathrm{2}}×\sqrt{\mathrm{2}}}{\cancel{\mathrm{2}}×\mathrm{4}}\:\Rightarrow\:\frac{\sqrt{\mathrm{6}}×\sqrt{\mathrm{2}}}{\mathrm{4}}\:\Rightarrow\:\frac{\sqrt{\mathrm{2}×\mathrm{6}}}{\mathrm{4}} \\ $$$$\:\Rightarrow\:\frac{\sqrt{\mathrm{2}×\mathrm{2}×\mathrm{3}}}{\mathrm{4}}\:\Rightarrow\:\frac{\sqrt{\mathrm{2}^{\mathrm{2}} ×\mathrm{3}}}{\mathrm{4}}\:\Rightarrow\:\frac{\sqrt{\mathrm{2}^{\mathrm{2}} }×\sqrt{\mathrm{3}}}{\mathrm{4}} \\ $$$$\:\Rightarrow\:\frac{\mathrm{2}×\sqrt{\mathrm{3}}}{\mathrm{4}}\:\Rightarrow\:\frac{\cancel{\mathrm{2}}\sqrt{\mathrm{3}}}{\cancel{\mathrm{2}}×\mathrm{2}}\:=\:\frac{\sqrt{\mathrm{3}}}{\mathrm{2}} \\ $$$$\:\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\:\approx\:\mathrm{0}.\mathrm{866025} \\ $$$$\:\sqrt{\frac{\mathrm{3000000}\:+\:\mathrm{3000000}}{\mathrm{4000000}\:+\:\mathrm{4000000}}}\:\approx\:\mathrm{0}.\mathrm{866025} \\ $$