Trong bài này ta tổng hợp các kĩ năng thực hiện các phép tính toán, khai căn, phân tích thành tích, trục căn thức ở mẫu để làm các bài toán phức tạp hơn.
Chú ý khi làm bài. Trong các bài này ta có thể rút gọn các phân thức riêng lẻ trước nếu được bằng cách phân tích thành tích, tiếp theo thì trục căn thức và rút gọn các biểu thức trong ngoặc, không nên qui đồng vì tính toán sẽ rất phức tạp.
Ví dụ 1. Rút gọn
a) $\dfrac{6-6\sqrt{3}}{1-\sqrt{3}}+\dfrac{3\sqrt{3}+3}{\sqrt{3}+1}$.
b) $\dfrac{2-\sqrt{2}}{1-\sqrt{2}}+\dfrac{\sqrt{2}-\sqrt{6}}{\sqrt{3}-1}$.
c) $\dfrac{\sqrt{15}-\sqrt{5}}{\sqrt{3}-1}+\dfrac{5-2\sqrt{5}}{2\sqrt{5}-4}$.
d) $\dfrac{3\sqrt{2}-6}{\sqrt{2}-1}+\dfrac{6\sqrt{2}-4}{\sqrt{2}-3}$.
Giải
a) $\dfrac{6-6\sqrt{3}}{1-\sqrt{3}}+\dfrac{3\sqrt{3}+3}{\sqrt{3}+1}$.\\
Ta có:\\
$\begin{aligned}
&\dfrac{6-6\sqrt{3}}{1-\sqrt{3}}+\dfrac{3\sqrt{3}+3}{\sqrt{3}+1}\\
&=\dfrac{6\left(1-\sqrt{3}\right)}{1-\sqrt{3}}+\dfrac{3\left(\sqrt{3}+1\right)}{\sqrt{3}+1}\\
&=6+3\\
&=9
\end{aligned}$
b) $\dfrac{2-\sqrt{2}}{1-\sqrt{2}}+\dfrac{\sqrt{2}-\sqrt{6}}{\sqrt{3}-1}$.\\
Ta có:\\
$\begin{aligned}
&\dfrac{2-\sqrt{2}}{1-\sqrt{2}}+\dfrac{\sqrt{2}-\sqrt{6}}{\sqrt{3}-1}\\
&=\dfrac{\sqrt{2}\left(\sqrt{2}-1\right)}{-\left(\sqrt{2}-1\right)}+\dfrac{\sqrt{2}\left(1-\sqrt{3}\right)}{-\left(1-\sqrt{3}\right)}\\
&=-\sqrt{2}-\sqrt{2}\\
&=-2\sqrt{2}
\end{aligned}$
c) $\dfrac{\sqrt{15}-\sqrt{5}}{\sqrt{3}-1}+\dfrac{5-2\sqrt{5}}{2\sqrt{5}-4}$.\\
Ta có:\\
$\begin{aligned}
&\dfrac{\sqrt{15}-\sqrt{5}}{\sqrt{3}-1}+\dfrac{5-2\sqrt{5}}{2\sqrt{5}-4}\\
&=\dfrac{\sqrt{5}\left(\sqrt{3}-1\right)}{\sqrt{3}-1}+\dfrac{\sqrt{5}\left(\sqrt{5}-2\right)}{2\left(\sqrt{5}-2\right)}\\
&=\sqrt{5}+\dfrac{\sqrt{5}}{2}\\
&=\dfrac{3\sqrt{5}}{2}
\end{aligned}$
d) $\dfrac{3\sqrt{2}-6}{\sqrt{2}-1}+\dfrac{6\sqrt{2}-4}{\sqrt{2}-3}$.\\
Ta có:\\
$\begin{aligned}
&\dfrac{3\sqrt{2}-6}{\sqrt{2}-1}+\dfrac{6\sqrt{2}-4}{\sqrt{2}-3}\\
&=\dfrac{3\sqrt{2}\left(1-\sqrt{2}\right)}{-\left(1-\sqrt{2}\right)}+\dfrac{2\sqrt{2}\left(3-\sqrt{2}\right)}{-\left(3-\sqrt{2}\right)}\\
&=-3\sqrt{2}-2\sqrt{2}\\
&=-5\sqrt{2}
\end{aligned}$
Ví dụ 2. Rút gọn
a) $\dfrac{6}{\sqrt{5}-1}+\dfrac{7}{1-\sqrt{3}}-\dfrac{2}{\sqrt{3}-\sqrt{5}}$.
b) $\dfrac{\sqrt{12}-6}{\sqrt{8}-\sqrt{24}}-\dfrac{3+\sqrt{3}}{\sqrt{3}}+\dfrac{4}{1-\sqrt{7}}$.
c) $\dfrac{1}{\sqrt{2}-\sqrt{3}}-\dfrac{1}{\sqrt{3}-\sqrt{5}}+\dfrac{1}{\sqrt{7}-\sqrt{5}}$.
d) $\left(\dfrac{\sqrt{14}-\sqrt{7}}{1-\sqrt{2}}+\dfrac{\sqrt{15}-\sqrt{5}}{1-\sqrt{3}}\right):\dfrac{1}{\sqrt{7}-\sqrt{5}}$.
Giải
a)$\dfrac{6}{\sqrt{5}-1}+\dfrac{7}{1-\sqrt{3}}-\dfrac{2}{\sqrt{3}-\sqrt{5}}$.
Ta có:
$\begin{aligned}
&\dfrac{6}{\sqrt{5}-1}+\dfrac{7}{1-\sqrt{3}}-\dfrac{2}{\sqrt{3}-\sqrt{5}}\\
&=\dfrac{6}{5-1}\left(\sqrt{5}+1\right)+\dfrac{7}{1-3}\left(1+\sqrt{3}\right)-\dfrac{2}{3-5}\left(\sqrt{3}+\sqrt{5}\right)\\
&=\dfrac{3}{2}\left(\sqrt{5}+1\right)-\dfrac{7}{2}\left(1+\sqrt{3}\right)+\sqrt{3}+\sqrt{5}\\
&=\dfrac{3\sqrt{5}}{2}+\dfrac{3}{2}-\dfrac{7}{2}-\dfrac{7\sqrt{3}}{2}+\sqrt{3}+\sqrt{5}\\
&=\dfrac{5\sqrt{5}}{2}-\dfrac{5\sqrt{3}}{2}-2\\
&=\dfrac{5}{2}\left(\sqrt{5}-\sqrt{3}\right)-2
\end{aligned}$
b) $\dfrac{\sqrt{12}-6}{\sqrt{8}-\sqrt{24}}-\dfrac{3+\sqrt{3}}{\sqrt{3}}+\dfrac{4}{1-\sqrt{7}}$.
Ta có:
$\begin{aligned}
&\dfrac{\sqrt{12}-6}{\sqrt{8}-\sqrt{24}}-\dfrac{3+\sqrt{3}}{\sqrt{3}}+\dfrac{4}{1-\sqrt{7}}\\
&=\dfrac{\sqrt{6}\left(\sqrt{2}-\sqrt{6}\right)}{2\left(\sqrt{2}-\sqrt{6}\right)}-\dfrac{\sqrt{3}\left(\sqrt{3}+1\right)}{\sqrt{3}}+\dfrac{4}{1-7}\left(1+\sqrt{7}\right)\\
&=\dfrac{\sqrt{6}}{2}-\left(\sqrt{3}+1\right)-\dfrac{2}{3}\left(1+\sqrt{7}\right)\\
&=-\dfrac{2}{3}\sqrt{7}+\dfrac{\sqrt{6}}{2}-\sqrt{3}-\dfrac{5}{3}
\end{aligned}$
c) $\dfrac{1}{\sqrt{2}-\sqrt{3}}-\dfrac{1}{\sqrt{3}-\sqrt{5}}+\dfrac{1}{\sqrt{7}-\sqrt{5}}$.\\
Ta có:\\
$\begin{aligned}
&\dfrac{1}{\sqrt{2}-\sqrt{3}}-\dfrac{1}{\sqrt{3}-\sqrt{5}}+\dfrac{1}{\sqrt{7}-\sqrt{5}}\\
&=\dfrac{1}{2-3}\left(\sqrt{2}+\sqrt{3}\right)-\dfrac{1}{3-5}\left(\sqrt{3}+\sqrt{5}\right)+\dfrac{1}{7-5}\left(\sqrt{7}+\sqrt{5}\right)\\
&=-\left(\sqrt{2}+\sqrt{3}\right)+\dfrac{1}{2}\left(\sqrt{3}+\sqrt{5}\right)+\dfrac{1}{2}\left(\sqrt{7}+\sqrt{5}\right)\\
&=-\sqrt{2}-\sqrt{3}+\dfrac{1}{2}\sqrt{3}+\dfrac{1}{2}\sqrt{5}+\dfrac{1}{2}\sqrt{7}+\dfrac{1}{2}\sqrt{5}\\
&=\dfrac{1}{2}\sqrt{7}+\sqrt{5}-\dfrac{1}{2}\sqrt{3}-\sqrt{2}
\end{aligned}$
d) $\left(\dfrac{\sqrt{14}-\sqrt{7}}{1-\sqrt{2}}+\dfrac{\sqrt{15}-\sqrt{5}}{1-\sqrt{3}}\right):\dfrac{1}{\sqrt{7}-\sqrt{5}}$.\\
Ta có:
$\begin{aligned}
&\left(\dfrac{\sqrt{14}-\sqrt{7}}{1-\sqrt{2}}+\dfrac{\sqrt{15}-\sqrt{5}}{1-\sqrt{3}}\right):\dfrac{1}{\sqrt{7}-\sqrt{5}}\\
&=\left[\dfrac{\sqrt{7}\left(\sqrt{2}-1\right)}{-\left(\sqrt{2}-1\right)}+\dfrac{\sqrt{5}\left(\sqrt{3}-1\right)}{-\left(\sqrt{3}-1\right)}\right].\left(\sqrt{7}-\sqrt{5}\right)\\
&=\left(-\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)\\
&=-(7-5)\\
&=-2
\end{aligned}$
Ví dụ 3. Rút gọn
a) $\left(\dfrac{12}{\sqrt{5}+1}-\dfrac{4}{\sqrt{5}+2}+\dfrac{20}{3+\sqrt{5}}\right)(10+3\sqrt{5})$.
b) $\left(\dfrac{24}{\sqrt{7}+1}+\dfrac{4}{3+\sqrt{7}}-\dfrac{3}{\sqrt{7}+2}\right)(4-\sqrt{7})$.
c) $\left(\dfrac{8}{\sqrt{3}-1}-\dfrac{4}{\sqrt{3}+1}+\dfrac{4}{\sqrt{5}+\sqrt{3}}\right):\sqrt{14+6\sqrt{5}}$.
d) $\left(\dfrac{7}{\sqrt{2}-1}+\dfrac{56}{\sqrt{2}-4}+\dfrac{3}{\sqrt{3}+\sqrt{2}}\right):\sqrt{12-6\sqrt{3}}$.
Giải
a) $\left(\dfrac{12}{\sqrt{5}+1}-\dfrac{4}{\sqrt{5}+2}+\dfrac{20}{3+\sqrt{5}}\right)(10+3\sqrt{5})$.
Ta có:
$\begin{aligned}
&\left(\dfrac{12}{\sqrt{5}+1}-\dfrac{4}{\sqrt{5}+2}+\dfrac{20}{3+\sqrt{5}}\right)(10+3\sqrt{5})\\
&=\left[\dfrac{12}{5-1}\left(\sqrt{5}-1\right)-\dfrac{4}{5-4}\left(\sqrt{5}-2\right)+\dfrac{20}{9-5}\left(3-\sqrt{5}\right)\right]\left(10+3\sqrt{5}\right)\\
&=\left[3\left(\sqrt{5}-1\right)-4\left(\sqrt{5}-2\right)+5\left(3-\sqrt{5}\right)\right]\left(10+3\sqrt{5}\right)\\
&=\left[3\sqrt{5}-3-4\sqrt{5}+8+15-5\sqrt{5}\right]\left(10+3\sqrt{5}\right)\\
&=\left(-6\sqrt{5}+20\right)\left(10+3\sqrt{5}\right)\\
&=2\left(10-3\sqrt{5}\right)\left(10+3\sqrt{5}\right)\\
&=2(100-45)\\
&=110
\end{aligned}$
b) $\left(\dfrac{24}{\sqrt{7}+1}+\dfrac{4}{3+\sqrt{7}}-\dfrac{3}{\sqrt{7}+2}\right)(4-\sqrt{7})$.
Ta có:
$\begin{aligned}
&\left(\dfrac{24}{\sqrt{7}+1}+\dfrac{4}{3+\sqrt{7}}-\dfrac{3}{\sqrt{7}+2}\right)(4-\sqrt{7})\\
&=\left[\dfrac{24}{7-1}\left(\sqrt{7}-1\right)+\dfrac{4}{9-7}\left(3-\sqrt{7}\right)-\dfrac{3}{7-4}\left(\sqrt{7}-2\right)\right]\left(4-\sqrt{7}\right)\\
&=\left[4\left(\sqrt{7}-1\right)+2\left(3-\sqrt{7}\right)-\left(\sqrt{7}-2\right)\right]\left(4-\sqrt{7}\right)\\
&=\left(4\sqrt{7}-4+6-2\sqrt{7}-\sqrt{7}+2\right)\left(4-\sqrt{7}\right)\\
&=\left(\sqrt{7}+4\right)\left(4-\sqrt{7}\right)\\
&=16-7
&=9
\end{aligned}$
c) $\left(\dfrac{8}{\sqrt{3}-1}-\dfrac{4}{\sqrt{3}+1}+\dfrac{4}{\sqrt{5}+\sqrt{3}}\right):\sqrt{14+6\sqrt{5}}$.
Ta có:
$\begin{aligned}
&\left(\dfrac{8}{\sqrt{3}-1}-\dfrac{4}{\sqrt{3}+1}+\dfrac{4}{\sqrt{5}+\sqrt{3}}\right):\sqrt{14+6\sqrt{5}}\\
&=\left[\dfrac{8}{3-1}\left(\sqrt{3}+1\right)-\dfrac{4}{3-1}\left(\sqrt{3}-1\right)+\dfrac{4}{5-3}\left(\sqrt{5}-\sqrt{3}\right)\right]:\left(3+\sqrt{5}\right)\\
&=\left[4\left(\sqrt{3}+1\right)-2\left(\sqrt{3}-1\right)+2\left(\sqrt{5}-\sqrt{3}\right)\right]:\left(3+\sqrt{5}\right)\\
&=\left(4\sqrt{3}+4-2\sqrt{3}+2+2\sqrt{5}-2\sqrt{3}\right):\left(3+\sqrt{5}\right)\\
&=\left(6+2\sqrt{5}\right):\left(3+\sqrt{5}\right)\\
&=2
\end{aligned}$
d) $\left(\dfrac{7}{\sqrt{2}-1}+\dfrac{56}{\sqrt{2}-4}+\dfrac{3}{\sqrt{3}+\sqrt{2}}\right):\sqrt{12-6\sqrt{3}}$.\\
Ta có:
$\begin{aligned}
&\left(\dfrac{7}{\sqrt{2}-1}+\dfrac{56}{\sqrt{2}-4}+\dfrac{3}{\sqrt{3}+\sqrt{2}}\right):\sqrt{12-6\sqrt{3}}\\
&=\left[\dfrac{7}{2-1}\left(\sqrt{2}+1\right)+\dfrac{56}{2-16}\left(\sqrt{2}+4\right)+\dfrac{3}{3-2}\left(\sqrt{3}-\sqrt{2}\right)\right]:\left(3-\sqrt{3}\right)\\
&=\left[7\left(\sqrt{2}+1\right)-4\left(\sqrt{2}+4\right)+3\left(\sqrt{3}-\sqrt{2}\right)\right]:\left(3-\sqrt{3}\right)\\
&=\left(7\sqrt{2}+7-4\sqrt{2}-16+3\sqrt{3}-3\sqrt{2}\right):\left(3-\sqrt{3}\right)\\
&=\left(-9+3\sqrt{3}\right):\left(3-\sqrt{3}\right)\\
&=-3
\end{aligned}$
Bài tập rèn luyện
Bài 1. Rút gọn
a) $\dfrac{\sqrt{160}-\sqrt{80}}{\sqrt{8}-\sqrt{2}}-\dfrac{\sqrt{40}-\sqrt{15}}{2\sqrt{2}-\sqrt{3}}$.
b) $\left(\dfrac{5-2\sqrt{5}}{2-\sqrt{5}}-2\right)\left(\dfrac{5+3\sqrt{5}}{3+\sqrt{5}}-2\right)$.
c) $\left(\dfrac{\sqrt{216}}{3}-\dfrac{2\sqrt{3}-\sqrt{6}}{\sqrt{8}-2}\right)\dfrac{1}{\sqrt{6}}$.
d) $\left(\dfrac{\sqrt{343}}{21}-\dfrac{28+4\sqrt{7}}{\sqrt{63}+3}\right)\dfrac{\sqrt{7}}{7}$.
Bài 2. Rút gọn
a) $\dfrac{5\sqrt{2}-2\sqrt{5}}{\sqrt{5}-\sqrt{2}}+\dfrac{6}{2-\sqrt{10}}$.
b) $\dfrac{3}{\sqrt{5}-\sqrt{2}}-\dfrac{2}{2-\sqrt{2}}+\dfrac{1}{\sqrt{3}+\sqrt{2}}$.
c) $\dfrac{-4}{\sqrt{7}-\sqrt{5}}+\dfrac{1}{\sqrt{3}-1}+\dfrac{4-2\sqrt{5}}{\sqrt{5}-2}$.
d) $\dfrac{5}{3-\sqrt{7}}-\dfrac{2}{\sqrt{2}+\sqrt{3}}+\dfrac{-1}{\sqrt{2}-1}$.
Bài 3. Rút gọn
a) $\dfrac{(\sqrt{3}-\sqrt{5})^2+4\sqrt{15}}{\sqrt{3}+\sqrt{5}}$.
b) $(\sqrt{5}+2)\dfrac{(\sqrt{5}+2)^2-8\sqrt{5}}{\sqrt{5}-2}$.
c) $\dfrac{(\sqrt{2}+1)^2-4\sqrt{2}}{\sqrt{2}-1}\cdot(\sqrt{2}+1)$.
d) $\dfrac{(\sqrt{3}-\sqrt{2})^2+4\sqrt{6}}{(\sqrt{3}+\sqrt{2})^2}\cdot(\sqrt{3}-\sqrt{2})$.
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