Question

Rút gọn (điều kiện các biểu thức có nghĩa)

a) \(\sqrt{(2-\sqrt{5)}^2}+\sqrt{14+6\sqrt{5}}\)

b) \((\sqrt{x^2-25})^2-(x-2)(x+3)\)

c) \(\sqrt{(2\sqrt{2}-7)^2}-\sqrt{44-24\sqrt{2}}\)

d) \((\sqrt{4-x^2})^2-(x-1)(3-x)\)

Answer 1

a) \(\sqrt{\left(2-\sqrt{5}\right)^2}+\sqrt{14+6\sqrt{5}}=\left|2-\sqrt{5}\right|+\sqrt{5+6\sqrt{5}+9}=\sqrt{5}-2+\sqrt{\left(\sqrt{5}+3\right)^2}=5-2+\sqrt{5}+3=2\sqrt{5}+1\)b) ĐK: \(\left[{}\begin{matrix}x\ge5\\x\le-5\end{matrix}\right.\)

\(\left(\sqrt{x^2-25}\right)^2-\left(x-2\right)\left(x+3\right)=\left(x^2-25\right)-\left(x^2-2x+3x-6\right)=x^2-25-x^2-x+6=-x-19\)

c) \(\sqrt{\left(2\sqrt{2}-7\right)^2}-\sqrt{44-24\sqrt{2}}=\left|2\sqrt{2}-7\right|-\sqrt{4\left(9-6\sqrt{2}+2\right)}=7-2\sqrt{2}-2\sqrt{\left(3-\sqrt{2}\right)^2}=7-2\sqrt{2}-2\left|3-\sqrt{2}\right|=7-2\sqrt{2}-6+2\sqrt{2}=1\)d) ĐK: \(-2\le x\le2\)

\(\left(\sqrt{4-x^2}\right)^2-\left(x-1\right)\left(3-x\right)=4-x^2-\left(3x-x^2+x-3\right)=4-x^2-4x+x^2+3=7-4x\)