Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
a) \(\frac{\sqrt{2}}{1+\sqrt{2}-\sqrt{3}}=\frac{\sqrt{2}.\left(1-\sqrt{2}+\sqrt{3}\right)}{\left(1+\sqrt{2}-\sqrt{3}\right).\left(1-\sqrt{2}+\sqrt{3}\right)}.\)
\(=\frac{\sqrt{2}.\left(1-\sqrt{2}+\sqrt{3}\right)}{1-\left(\sqrt{2}-\sqrt{3}\right)^2}=\frac{\sqrt{2}.\left(1-\sqrt{2}+\sqrt{3}\right)}{1-\left(5-2\sqrt{6}\right)}\)
\(=\frac{\sqrt{2}.\left(1-\sqrt{2}+\sqrt{3}\right)}{-4+2\sqrt{6}}=\frac{1-\sqrt{2}+\sqrt{3}}{-2\sqrt{2}+2\sqrt{3}}\)
\(=\frac{\left(1-\sqrt{2}+\sqrt{3}\right).\left(\sqrt{2}+\sqrt{3}\right)}{-2\left(\sqrt{2}-\sqrt{3}\right).\left(\sqrt{2}+\sqrt{3}\right)}\)
\(=\frac{\left(1-\sqrt{2}+\sqrt{3}\right).\left(\sqrt{2}+\sqrt{3}\right)}{-2.\left(2-3\right)}\)\(=\frac{\left(1-\sqrt{2}+\sqrt{3}\right).\left(\sqrt{2}+\sqrt{3}\right)}{2}\)
Căn thức ở mẫu đã được trục rồi.
Nếu cần thì phá ngoặc phần tử số ra.
b) Nhân cả tử số và mẫu số cho \(\sqrt{a+3}-\sqrt{a-3}\)thì mẫu số có giá trị là (a + 3) - (a - 3) = 6; tử số có giá trị là \(\left(\sqrt{a+3}-\sqrt{a-3}\right)^2\). Khi đó, căn thức ở mẫu đã được trục đi rồi. Sau đó bạn phá ngoặc phần tử số ra.
a/ \(\frac{1}{2+\sqrt{3}}-\frac{1}{2-\sqrt{3}}+5\sqrt{3}\)
\(=\frac{2-\sqrt{3}}{\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)}-\frac{2+\sqrt{3}}{\left(2-\sqrt{3}\right)\left(2+\sqrt{3}\right)}+5\sqrt{3}\)
\(=\frac{2-\sqrt{3}}{4-3}-\frac{2+\sqrt{3}}{4-3}+5\sqrt{3}\)
\(=2-\sqrt{3}-2-\sqrt{3}+5\sqrt{3}\)
\(=3\sqrt{3}\)
Vậy..
b/ \(\frac{1}{\sqrt{5}+2}-\sqrt{9+4\sqrt{5}}\)
\(=\frac{1}{\sqrt{5}+2}-\sqrt{\left(\sqrt{5}+2\right)^2}\)
\(=\frac{1}{\sqrt{5}+2}-\left|\sqrt{5}+2\right|\)
\(=\frac{\sqrt{5}-2}{\left(\sqrt{5}-2\right)\left(\sqrt{5}+2\right)}-\sqrt{5}-2\)
\(=\sqrt{5}-2-\sqrt{5}-2\)
\(=-4\)
Vậy..
\(a,\frac{1}{\sqrt{2}+\sqrt{3}-\sqrt{6}}=\frac{\sqrt{2}+\sqrt{3}+\sqrt{6}}{\left(\sqrt{2}+\sqrt{3}-\sqrt{6}\right)\left(\sqrt{2}+\sqrt{3}+\sqrt{6}\right)}=\frac{\sqrt{2}+\sqrt{3}+\sqrt{6}}{\left(\sqrt{2}+\sqrt{3}\right)^2-\sqrt{6}^2}\)
\(=\frac{\sqrt{2}+\sqrt{3}+\sqrt{6}}{2\sqrt{6}-1}=\frac{\left(\sqrt{2}+\sqrt{3}+\sqrt{6}\right)\left(2\sqrt{6}+1\right)}{2\sqrt{6}^2-1^2}=\frac{4\sqrt{3}+6\sqrt{2}+12+\sqrt{2}+\sqrt{3}+\sqrt{6}}{11}\)\(=\frac{\sqrt{6}+5\sqrt{3}+7\sqrt{2}+12}{11}\)
\(b,\frac{1}{\sqrt{x}+\sqrt{y}+\sqrt{z}}=\frac{\sqrt{x}+\sqrt{y}-\sqrt{z}}{\left(\sqrt{z}+\sqrt{y}+\sqrt{z}\right)\left(\sqrt{x}+\sqrt{y}-\sqrt{z}\right)}=\frac{\sqrt{x}+\sqrt{y}-\sqrt{z}}{\left(\sqrt{x}+\sqrt{y}\right)^2-\sqrt{z}^2}\)
\(=\frac{\sqrt{x}+\sqrt{y}-\sqrt{z}}{x+2\sqrt{xy}+y-z}\)
\(A=\frac{\sqrt{a+b}-\sqrt{a}+\sqrt{b}}{\sqrt{a+b}-\sqrt{a}-\sqrt{b}}=1+\frac{2\sqrt{b}}{\sqrt{a+b}-\sqrt{a}-\sqrt{b}}=1+B\)
\(B=\frac{2\sqrt{b}\left(\sqrt{a+b}+\sqrt{a}+\sqrt{b}\right)}{-2\sqrt{ab}}=-\frac{\left(\sqrt{a+b}+\sqrt{a}+\sqrt{b}\right)}{\sqrt{a}}=-\frac{\sqrt{a}\left(\sqrt{a+b}+\sqrt{a}+\sqrt{b}\right)}{a}\)
\(=\frac{\left(\sqrt{a+b}-\sqrt{a}+\sqrt{b}\right)^2}{\left(\sqrt{a+b}-\sqrt{a}\right)^2-b}\)
\(=\frac{2a+2b-2\sqrt{a\left(a+b\right)}+2\sqrt{b\left(a+b\right)}-2\sqrt{ab}}{2a-2\sqrt{a\left(a+b\right)}}\)
\(=\frac{a+b-\sqrt{a\left(a+b\right)}+\sqrt{b\left(a+b\right)}-\sqrt{ab}}{a-\sqrt{a\left(a+b\right)}}\)
\(=\frac{\left(a+b-\sqrt{a\left(a+b\right)}+\sqrt{b\left(a+b\right)}-\sqrt{ab}\right)\left(a+\sqrt{a\left(a+b\right)}\right)}{a^2-a\left(a+b\right)}\)
\(=\frac{b\sqrt{a\left(a+b\right)}+\sqrt{ab}\left(a+b\right)-a\sqrt{ab}}{-ab}\)
\(=\frac{-b\sqrt{a\left(a+b\right)}+b\sqrt{ab}}{ab}\)
\(=\frac{\sqrt{ab}-\sqrt{a\left(a+b\right)}}{a}\)
\(\frac{\left(\sqrt{a}\right)^3-\left(\sqrt{b}\right)^3}{\sqrt{a}-\sqrt{b}}=\frac{\left(\sqrt{a}-\sqrt{b}\right)\left(a+\sqrt{ab}+b\right)}{\sqrt{a}-\sqrt{b}}\\ \)
\(a+\sqrt{ab}+b\)
Ta có:
\(\frac{a\sqrt{a}-b\sqrt{b}}{\sqrt{a}-\sqrt{b}}\)
\(\Leftrightarrow\frac{\sqrt{a}^3-\sqrt{b}^3}{\sqrt{a}-\sqrt{b}}\)
\(\Leftrightarrow\frac{\left(\sqrt{a}-\sqrt{b}\right)\left(a+\sqrt{ab}+b\right)}{\sqrt{a}-\sqrt{b}}\)
\(\Rightarrow a+\sqrt{ab}+b\)
\(a,\frac{\sqrt{5}}{\sqrt{3-\sqrt{5}}}=\frac{\sqrt{5}\left(\sqrt{3+\sqrt{5}}\right)}{\sqrt{\left(3-\sqrt{5}\right).\left(3+\sqrt{5}\right)}}\)
\(=\frac{\sqrt{5}\left(\sqrt{3+\sqrt{5}}\right)}{\sqrt{9-5}}=\frac{\sqrt{5}\left(\sqrt{3+\sqrt{5}}\right)}{\sqrt{4}}=\frac{\sqrt{5}\left(\sqrt{3+\sqrt{5}}\right)}{2}\)
a) \(\frac{a-\sqrt{a}}{\sqrt{a}-1}=\frac{\left(a-\sqrt{a}\right)\left(\sqrt{a}+1\right)}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}=\frac{a\sqrt{a}-\sqrt{a}}{a-1}=\frac{\sqrt{a}\left(a-1\right)}{a-1}=\sqrt{a}\)
b) \(\frac{p-2\sqrt{p}}{p-\sqrt{2}}=\frac{\left(p-2\sqrt{p}\right)\left(p+\sqrt{2}\right)}{\left(p-\sqrt{2}\right)\left(p+\sqrt{2}\right)}=\frac{\left(p-2\sqrt{p}\right)\left(p+\sqrt{2}\right)}{p^2-2}\)