Question

Câu 3/

Cho biểu thức p=\(P=\left(\frac{1}{1-\sqrt{y}}+\frac{1}{1+\sqrt{y}}\right):\left(\frac{1}{1-\sqrt{y}}-\frac{1}{1+\sqrt{y}}\right)+\frac{1}{1-\sqrt{y}}\)

a) Tìm ĐKXĐ và rút gọn biểu thức P

b) Tính giá trị của P khi y=\(4+2\sqrt{3}\)

Answer 1

ĐKXĐ: \(y>0;y\ne1\)

\(P=\left(\frac{1+\sqrt{y}+1-\sqrt{y}}{\left(1-\sqrt{y}\right)\left(1+\sqrt{y}\right)}\right):\left(\frac{1+\sqrt{y}-\left(1-\sqrt{y}\right)}{\left(1-\sqrt{y}\right)\left(1+\sqrt{y}\right)}\right)+\frac{1}{1-\sqrt{y}}\)

\(=\frac{2}{\left(1-\sqrt{y}\right)\left(1+\sqrt{y}\right)}:\frac{2\sqrt{y}}{\left(1-\sqrt{y}\right)\left(1+\sqrt{y}\right)}+\frac{1}{1-\sqrt{y}}\)

\(=\frac{1}{\sqrt{y}}+\frac{1}{1-\sqrt{y}}=\frac{1}{\sqrt{y}\left(1-\sqrt{y}\right)}\)

\(y=4+2\sqrt{3}=\left(\sqrt{3}+1\right)^2\Rightarrow\sqrt{y}=\sqrt{3}+1\)

\(P=\frac{1}{\left(\sqrt{3}+1\right)\left(1-\sqrt{3}-1\right)}=\frac{-1}{3+\sqrt{3}}\)