Giải bài tập 3.28 trang 64

Đề bài

Rút gọn các biểu thức sau:

a) $\frac{5+3\sqrt{5}}{\sqrt{5}}-\frac{1}{\sqrt{5}-2};$

b) $\sqrt{{\left(\sqrt{7}-2\right)}^2}-\sqrt{63}+\frac{\sqrt{56}}{\sqrt{2}};$

c) $\frac{\sqrt{{\left(\sqrt{3}+\sqrt{2}\right)}^2}+\sqrt{{\left(\sqrt{3}-\sqrt{2}\right)}^2}}{2\sqrt{12}};$

d) $\frac{\sqrt[3]{3{\left(\sqrt{2}+1\right)}^3}-1}{\sqrt{50}}.$

Lời giải chi tiết

a) $\frac{5+3\sqrt{5}}{\sqrt{5}}-\frac{1}{\sqrt{5}-2}$

$\begin{array}{c}=\frac{\sqrt{5}\left(\sqrt{5}+3\right)}{\sqrt{5}}-\frac{\sqrt{5}+2}{\left(\sqrt{5}-2\right)\left(\sqrt{5}+2\right)}\\ =\sqrt{5}+3-\frac{\sqrt{5}+2}{5-4}\end{array}$

$\begin{array}{c}=\sqrt{5}+3-\left(\sqrt{5}+2\right)\\ =1\end{array}$

b) $\sqrt{{\left(\sqrt{7}-2\right)}^2}-\sqrt{63}+\frac{\sqrt{56}}{\sqrt{2}}$

$\begin{array}{c}=\left|\sqrt{7}-2\right|-\sqrt{9.7}+\frac{\sqrt{2.28}}{\sqrt{2}}\\ =\sqrt{7}-2-3\sqrt{7}+\sqrt{28}\\ =-2-2\sqrt{7}+\sqrt{4.7}\end{array}$

$\begin{array}{c}=-2-2\sqrt{7}+2\sqrt{7}\\ =-2\end{array}$

c) $\frac{\sqrt{{\left(\sqrt{3}+\sqrt{2}\right)}^2}+\sqrt{{\left(\sqrt{3}-\sqrt{2}\right)}^2}}{2\sqrt{12}}$

$\begin{array}{c}=\frac{\left|\sqrt{3}+\sqrt{2}\right|+\left|\sqrt{3}-\sqrt{2}\right|}{2\sqrt{4.3}}\\ =\frac{\sqrt{3}+\sqrt{2}+\sqrt{3}-\sqrt{2}}{4\sqrt{3}}\\ =\frac{2\sqrt{3}}{4\sqrt{3}}\\ =\frac{1}{2}\end{array}$

d) $\frac{\sqrt[3]{3{\left(\sqrt{2}+1\right)}^3}-1}{\sqrt{50}}$

$\begin{array}{c}=\frac{\sqrt{2}+1-1}{\sqrt{25.2}}\\ =\frac{\sqrt{2}}{5\sqrt{2}}\\ =\frac{1}{5}\end{array}$