Question

Rút gọn biểu thức:

a) \(\sqrt{\sqrt{x^4+4}-x^2}.\sqrt{\sqrt{x^4+4}+x^2}\)

b) \(\dfrac{x\sqrt{y}+y\sqrt{x}}{x+2\sqrt{xy}+y}\) với x, y ≥ 0; xy ≠ 0

c) \(\dfrac{3\sqrt{a}-2a-1}{4a-4\sqrt{a}+1}\) với a ≥ 0, a ≠ 1/4

d) \(\dfrac{a+4\sqrt{a}+4}{\sqrt{a}+2}+\dfrac{4-a}{\sqrt{a}-2}\) với a ≥ = 0, a ≠ 4

Answer 1

`a)`\(\sqrt{\sqrt{x^4+4}-x^2}.\sqrt{\sqrt{x^4+4}+x^2}\)

\(=\sqrt{\left(\sqrt{x^4+4}-x^2\right).\left(\sqrt{x^4+4}+x^2\right)}\)

\(=\sqrt{\left(\sqrt{x^4+4}\right)^2-\left(x^2\right)^2}\)

\(=\sqrt{x^4+4-x^4}\)

\(=\sqrt{4}=2\)

`b)`\(\dfrac{x\sqrt{y}+y\sqrt{x}}{x+2\sqrt{xy}+y}=\dfrac{\sqrt{xy}\left(\sqrt{x}+\sqrt{y}\right)}{\left(\sqrt{x}+\sqrt{y}\right)^2}=\dfrac{\sqrt{xy}}{\sqrt{x}+\sqrt{y}}\)

`c)`\(\dfrac{3\sqrt{a}-2a-1}{4a-4\sqrt{a}+1}=\dfrac{\left(\sqrt{a}-1\right)\left(2\sqrt{a}-1\right)}{\left(2\sqrt{a}-1\right)^2}=\dfrac{\sqrt{a}-1}{2\sqrt{a}-1}\)

`d)`\(\dfrac{a+4\sqrt{a}+4}{\sqrt{a}+2}+\dfrac{4-a}{\sqrt{a}-2}\)

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

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

\(=0\)