Question

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Answer 1

d) \(\dfrac{3x}{7y}\cdot\sqrt{\dfrac{49y^2}{9x^2}}\)

\(=\dfrac{3x}{7y}\cdot\sqrt{\dfrac{\left(7y\right)^2}{\left(3x\right)^2}}\)

\(=\dfrac{3x}{7y}\cdot\dfrac{7y}{3x}\)

\(=\dfrac{21xy}{21xy}\)

\(=1\)

Answer 2

e) \(\sqrt{1+\dfrac{1}{a^2}+\dfrac{1}{\left(a+1\right)^2}}\)

\(=\sqrt{1+\dfrac{1}{a^2}+\dfrac{1}{\left(a+1\right)^2}+\dfrac{2\left(a+1-a-1\right)}{a\left(a+1\right)}}\)

\(=\sqrt{1^2+\left(\dfrac{1}{a}\right)^2+\left(\dfrac{1}{a+1}\right)^2+2\cdot1\cdot\dfrac{1}{a}-2\cdot1\cdot\dfrac{1}{a+1}-2\cdot\dfrac{1}{a}\cdot\dfrac{1}{a+1}}\)

\(=\sqrt{\left(1+\dfrac{1}{a}-\dfrac{1}{a+1}\right)^2}\)

\(=1+\dfrac{1}{a}-\dfrac{1}{a+1}\)