a) \(\dfrac{\sqrt{2}}{\sqrt{18}}=\sqrt{\dfrac{2}{18}}=\sqrt{\dfrac{2.1}{2.9}}=\sqrt{\dfrac{1}{9}}=\sqrt {{{\left( {\dfrac{1}{3}} \right)}^2}} =\dfrac{1}{3}\).
b) \(\dfrac{\sqrt{15}}{\sqrt{735}}=\sqrt{\dfrac{15}{735}}=\sqrt{\dfrac{15.1}{15.49}}=\sqrt{\dfrac{1}{49}}=\sqrt {{{\left( {\dfrac{1}{7}} \right)}^2}}\)
\(=\dfrac{1}{7}\).
c) \(\dfrac{\sqrt{12500}}{\sqrt{500}}=\sqrt{\dfrac{12500}{500}}=\sqrt{\dfrac{500.25}{500}}\)
\(=\sqrt{25}=\sqrt{5^2}=5\).
d) \(\dfrac{\sqrt{6^{5}}}{\sqrt{2^{3}.3^{5}}}=\sqrt{\dfrac{6^5}{2^3.3^5}}=\sqrt{\dfrac{(2.3)^5}{2^3.3^5}}=\sqrt{\dfrac{2^5.3^5}{2^3.3^5}}\)
\(=\sqrt{\dfrac{2^5.3^5}{2^3.3^5}}=\sqrt{\dfrac{2^5}{2^3}}=\sqrt{\dfrac{2^3.2^2}{2^3}}=\sqrt{2^2}=2\)
