Question

Answer 1

a) \(A=2\sqrt{2-x}-\dfrac{5}{2}\sqrt{8-4x}+\sqrt{32-16x}\)

\(A=2\sqrt{2-x}-\dfrac{5}{2}\cdot2\sqrt{2-x}+4\sqrt{2-x}\)

\(A=2\sqrt{2-x}-5\sqrt{2-x}+4\sqrt{2-x}\)

\(A=\sqrt{2-x}\)

b) \(\dfrac{x+\sqrt{xy}}{y+\sqrt{xy}}\)

\(=\dfrac{\sqrt{x}\left(\sqrt{x}+\sqrt{y}\right)}{\sqrt{y}\left(\sqrt{x}+\sqrt{y}\right)}\)

\(=\dfrac{\sqrt{x}}{\sqrt{y}}\)

c) \(C=\dfrac{\sqrt{x}+1}{\sqrt{x}-2}+\dfrac{2\sqrt{x}}{\sqrt{x}+2}+\dfrac{2+5\sqrt{x}}{4-x}\)

\(C=\dfrac{\sqrt{x}+1}{\sqrt{x}-2}+\dfrac{2\sqrt{x}}{\sqrt{x}+2}-\dfrac{2+5\sqrt{x}}{x-4}\)

\(C=\dfrac{\left(\sqrt{x}+1\right)\left(\sqrt{x}+2\right)}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}+\dfrac{2\sqrt{x}\left(\sqrt{x}-2\right)}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}-\dfrac{2+5\sqrt{x}}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}\)

\(C=\dfrac{x+2\sqrt{x}+\sqrt{x}+2+2x-4\sqrt{x}-2-5\sqrt{x}}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)

\(C=\dfrac{3x-6\sqrt{x}}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)

\(C=\dfrac{3\sqrt{x}\left(\sqrt{x}-2\right)}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}\)

\(C=\dfrac{3\sqrt{x}}{\sqrt{x}+2}\)

Answer 2

d) \(D=\left(\dfrac{3x+3}{x-9}-\dfrac{2\sqrt{x}+\sqrt{x}}{\sqrt{x}+3}+\dfrac{\sqrt{x}}{3-\sqrt{x}}\right):\dfrac{\sqrt{x}+1}{\sqrt{x}-3}\)

\(D=\left[\dfrac{3x+3}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}-\dfrac{2\sqrt{x}\left(\sqrt{x}-3\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}-\dfrac{\sqrt{x}\left(\sqrt{x}+3\right)}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\right]\cdot\dfrac{\sqrt{x}-3}{\sqrt{x}+1}\)

\(D=\dfrac{3x+3-2x+6\sqrt{x}-x-3\sqrt{x}}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}\cdot\dfrac{\sqrt{x}-3}{\sqrt{x}+1}\)

\(D=\dfrac{3\sqrt{x}+3}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}\cdot\dfrac{\sqrt{x}-3}{\sqrt{x}+1}\)

\(D=\dfrac{3\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}\cdot\dfrac{\sqrt{x}-3}{\sqrt{x}+1}\)

\(D=\dfrac{3\left(\sqrt{x}+1\right)\left(\sqrt{x}-3\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)\left(\sqrt{x}+1\right)}\)

\(D=\dfrac{3}{\sqrt{x}+3}\)