d) \(\dfrac{3x}{7y}\cdot\sqrt{\dfrac{49y^2}{9x^2}}\)
\(=\dfrac{3x}{7y}\cdot\sqrt{\dfrac{\left(7y\right)^2}{\left(3x\right)^2}}\)
\(=\dfrac{3x}{7y}\cdot\dfrac{7y}{3x}\)
\(=\dfrac{21xy}{21xy}\)
\(=1\)
e) \(\sqrt{1+\dfrac{1}{a^2}+\dfrac{1}{\left(a+1\right)^2}}\)
\(=\sqrt{1+\dfrac{1}{a^2}+\dfrac{1}{\left(a+1\right)^2}+\dfrac{2\left(a+1-a-1\right)}{a\left(a+1\right)}}\)
\(=\sqrt{1^2+\left(\dfrac{1}{a}\right)^2+\left(\dfrac{1}{a+1}\right)^2+2\cdot1\cdot\dfrac{1}{a}-2\cdot1\cdot\dfrac{1}{a+1}-2\cdot\dfrac{1}{a}\cdot\dfrac{1}{a+1}}\)
\(=\sqrt{\left(1+\dfrac{1}{a}-\dfrac{1}{a+1}\right)^2}\)
\(=1+\dfrac{1}{a}-\dfrac{1}{a+1}\)
