Rút gọn
\(\frac{\sqrt{108x^3}}{\sqrt{12x}}\) (x>0)
\(\frac{\sqrt{13x^4y^6}}{\sqrt{208x^6y^6}}\left(x< 0;y\ne0\right)\)
Rút gọn các biểu thức sau:
a)\(\frac{\sqrt{108x^3}}{\sqrt{12x}}\left(x>0\right)\)
b)\(\frac{\sqrt{13x^4y^6}}{\sqrt{208x^6y^6}}\left(x< 0;y\ne0\right)\)
c)\(\frac{x\sqrt{x}+y\sqrt{y}}{\sqrt{x}+\sqrt{y}}-\left(\sqrt{x}+\sqrt{y}\right)^2\)
d) \(\sqrt{\frac{x-2\sqrt{x}+1}{x+2\sqrt{x}+1}}\left(x\ge\right)\)
e)\(\frac{x-1}{\sqrt{y}-1}.\sqrt{\frac{\left(y-2\sqrt{y}+1\right)^2}{\left(x-1\right)^4}}\left(y>0;x\ne1;y\ne1\right)\)
\(a,\frac{\sqrt{108x^3}}{\sqrt{12x}}=\frac{\sqrt{36.3.x^3}}{\sqrt{3.4.x}}=\frac{6\sqrt{3}.\sqrt{x}^3}{2\sqrt{3}.\sqrt{x}}=3\sqrt{x}^2=3x\)
\(b,\frac{\sqrt{13x^4y^6}}{\sqrt{208x^6y^6}}=\frac{\sqrt{13}.\sqrt{x^4}.\sqrt{y^6}}{\sqrt{16.13}.\sqrt{x^6}.\sqrt{y^6}}=\frac{\sqrt{13}.x^2y^3}{4\sqrt{13}x^3y^3}=\frac{1}{4x}\)
\(c,\frac{x\sqrt{x}+y\sqrt{y}}{\sqrt{x}+\sqrt{y}}-\left(\sqrt{x}+\sqrt{y}\right)^2\)
\(=\frac{\sqrt{x}^3+\sqrt{y}^3}{\sqrt{x}+\sqrt{y}}-\left(x+2\sqrt{xy}+y\right)\)
\(=\frac{\left(\sqrt{x}+\sqrt{y}\right)\left(x-\sqrt{xy}+y\right)}{\sqrt{x}+\sqrt{y}}-x-2\sqrt{xy}-y\)
\(=x-\sqrt{xy}+y-x-2\sqrt{xy}-y=-3\sqrt{xy}\)
\(d,\sqrt{\frac{x-2\sqrt{x}+1}{x+2\sqrt{x}+1}}=\frac{\sqrt{\left(\sqrt{x}-1\right)^2}}{\sqrt{\left(\sqrt{x}+1\right)^2}}=\frac{\sqrt{x}-1}{\sqrt{x}+1}\)
Đk chỗ này là \(\sqrt{x}-1\ge0\Rightarrow\sqrt{x}\ge\sqrt{1}\Rightarrow x\ge1\)nhé
\(e,\frac{x-1}{\sqrt{y}-1}.\sqrt{\frac{\left(y-2\sqrt{y}+1\right)^2}{\left(x-1\right)^4}}=\frac{x-1}{\sqrt{y}-1}.\frac{y-2\sqrt{y}+1}{\left(x-1\right)^2}\)
\(=\frac{\left(x-1\right)\left(\sqrt{y}-1\right)^2}{\left(\sqrt{y}-1\right)\left(x-1\right)^2}=\frac{\sqrt{y}-1}{x-1}\)
Linh ơi, câu a,b,c bạn làm đều đúng hết kết quả cách làm đều đúng nhưng mà ở chỗ câu c): \(\sqrt{x}^3+\sqrt{y}^3\)
không phải vậy đâu, mặc dù mình biết bạn hiểu, hay do sơ suất, nhưng mà chỗ đó là \(\sqrt{x^3}+\sqrt{y^3}\)nha! Dù sao cũng cảm ơn bạn nha!
Rút gọn
a) \(\frac{x}{y}\sqrt{\frac{y^2}{x^4}}\left(x\ne0;y>0\right)\) b) \(3x^2\sqrt{\frac{8}{x^2}}\left(x< 0\right)\) c) \(2x^3y^3\sqrt{\frac{4}{x^8y^6}}\left(x\ne0;y< 0\right)\)
d)\(\frac{\sqrt{4x^4y^6}}{\sqrt{196x^6y^6}}\left(x< 0;y\ne0\right)\)
a. Ta có:\(\frac{x}{y}\sqrt{\frac{y^2}{x^4}=}\) \(\frac{x}{y}.\frac{\left|y\right|}{x^2}=\frac{x.y}{x^2y}\)\(=\frac{1}{x}\)(Vì \(x\ne0;y>0\))
b \(3x^2\sqrt{\frac{8}{x^2}}=3x^2\frac{2\sqrt{2}}{\left|x\right|}=\frac{6x^2\sqrt{2}}{-x}=-6x\sqrt{2}\)( Vì \(x< 0\))
bài 1: rút gọn các biểu thức.
a) \(\dfrac{x\sqrt{x}+y\sqrt{y}}{\sqrt{x}+\sqrt{y}}-(\sqrt{x}-\sqrt{y})^2\)
b) \(\sqrt{\dfrac{x-2\sqrt{x}+1}{x+2\sqrt{x}+1}}(x\ge0)\)
c) \(\dfrac{x-1}{\sqrt{y}-1}\sqrt{\dfrac{(y-2\sqrt{y}+1)^2}{(x-1)^4}}(x\ne1,y\ne1,y>0)\)
bài 2:rút gọn và tính.
a) \(\sqrt{\dfrac{\sqrt{a}-1}{\sqrt{b}+1}:}\sqrt{\dfrac{\sqrt{b}-1}{\sqrt{a}+1}với}a=7,25;b=3,25\)
b) \(\sqrt{15a^2-8a\sqrt{15}+16}vớia=\sqrt{\dfrac{3}{5}}+\sqrt{\dfrac{5}{3}}\)
c) \(\sqrt{10a^2-4a\sqrt{10}+4}vớia=\sqrt{\dfrac{2}{5}}+\sqrt{\dfrac{5}{2}}\)
d) \(\sqrt{a^2+2\sqrt{a^2-1}}-\sqrt{a^2-2\sqrt{a^2-1}}(a=\sqrt{5})\)
bài 3: rút gọn các biểu thức.
a) \(\sqrt{9(x-5)^2}(x\ge5)\)
b) \(\sqrt{x^2.(x-2)^2}(x< 0)\)
c)\(\dfrac{\sqrt{108x^3}}{\sqrt{12x}}(x>0)\)
d)\(\dfrac{\sqrt{13x^4y^6}}{\sqrt{208x^6y^6}}(x< 0:y\ne0)\)
ai giúp mik vs ạ, cảm ơn !
Bài 1:
a. ta có \(\dfrac{x\sqrt{x}+y\sqrt{y}}{\sqrt{x}+\sqrt{y}}-\left(\sqrt{x}-\sqrt{y}\right)^2\)
= \(\dfrac{\left(\sqrt{x}+\sqrt{y}\right)\left(x-\sqrt{xy}+y\right)}{\sqrt{x}+\sqrt{y}}-x+2\sqrt{xy}-y\)
= \(x-\sqrt{xy}+y-x+2\sqrt{xy}-y\)
=\(\sqrt{xy}\)
b.ĐK: x ≠ 1
Ta có: A= \(\sqrt{\dfrac{x+2\sqrt{x}+1}{x-2\sqrt{x}+1}}\)=\(\sqrt{\dfrac{\left(\sqrt{x}+1\right)^2}{\left(\sqrt{x}-1\right)^2}}\)=\(\dfrac{\sqrt{x}+1}{\left|\sqrt{x}-1\right|}\)
*Nếu \(\sqrt{x}-1\ge0\Rightarrow\sqrt{x}\ge1\)
⇒ A = \(\dfrac{\sqrt{x}+1}{\sqrt{x}-1}\)
*Nếu \(\sqrt{x}-1< 0\Rightarrow\sqrt{x}< 1\)
⇒ A=\(\dfrac{\sqrt{x}+1}{-\sqrt{x}+1}\)
c.Ta có:
a) \(Q=\frac{\left(\sqrt{x}-\sqrt{y}\right)^2+2x\sqrt{x}+y\sqrt{y}}{x\sqrt{x}+y\sqrt{y}}\left(x>0,y>0\right)\)
Rút Gọn
b) \(M=\frac{x^2-\sqrt{2}}{x^4+\left(\sqrt{3}-\sqrt{2}\right)x^2-\sqrt{6}}\)
Rút Gọn
1) cho biểu thức P=\(\frac{\sqrt{a}+2}{\sqrt{a}+3-}-\frac{5}{a+\sqrt{a}-6}+\frac{1}{2-\sqrt{a}}\)
a/ rút gọn P
b/ tìm giá trị của a để P<1
2) cho biểu thức P=\(\left(1-\frac{\sqrt{x}}{\sqrt{x}+1}\right):\left(\frac{\sqrt{x}+3}{\sqrt{x}-2}+\frac{\sqrt{x}+2}{3-\sqrt{x}}+\frac{\sqrt{x}+2}{x-5\sqrt{x}+6}\right)\)
a/ rút gọn P
b/ tìm giá trị của P<0
Rút gọn biểu thức sau
a/ A=\(\frac{x\sqrt{y}-y\sqrt{x}}{\sqrt{xy}}+\frac{x-y}{\sqrt{x}-\sqrt{y}}\)Với x>0 ; y>0 ;x#y
b/ B=\(\frac{3}{2+\sqrt{3}}+\frac{13}{4-\sqrt{3}}+\frac{6}{\sqrt{3}}\)
c/ C=\(\frac{\sqrt{4-2\sqrt{3}}}{\sqrt{6}-\sqrt{2}}\)
d/ D=\(\left(3\sqrt{2}+\sqrt{6}\right)\sqrt{6-3\sqrt{3}}\)
rút gọn giúp mình nha
Q=\(\frac{x-y}{\sqrt{x}-\sqrt{y}}\)\(-\frac{\sqrt{x^3}-\sqrt{y^3}}{x-y}\)với x ≥ 0, y ≥ 0 và x 6= y.
R=\(\left(\frac{1-a\sqrt{a}}{1-\sqrt{a}}+\sqrt{a}\right)\)\(\frac{\left(1-\sqrt{a}\right)^2}{\left(1-a\right)^2}\)với a ≥ 0 và a 6= 1.
\(Q=\frac{x-y}{\sqrt{x}-\sqrt{y}}-\frac{\sqrt{x^3}-\sqrt{y^3}}{x-y}\)
\(Q=\frac{\left(\sqrt{x}+\sqrt{y}\right)\left(x-y\right)-x\sqrt{x}+y\sqrt{y}}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}\)
\(Q=\frac{x\sqrt{x}-y\sqrt{x}+x\sqrt{y}-y\sqrt{y}-x\sqrt{x}+y\sqrt{y}}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}\)
\(Q=\frac{\sqrt{xy}\left(\sqrt{x}-\sqrt{y}\right)}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}\)
\(Q=\frac{\sqrt{xy}}{\sqrt{x}+\sqrt{y}}\)
\(R=\left(\frac{1-a\sqrt{a}}{1-\sqrt{a}}+\sqrt{a}\right).\frac{\left(1-\sqrt{a}\right)^2}{\left(1-a\right)^2}\)
\(R=\left[\frac{\left(1-\sqrt{a}\right)\left(1+\sqrt{a}+a\right)}{1-\sqrt{a}}+\sqrt{a}\right].\frac{\left(1-\sqrt{a}\right)^2}{\left(1-a\right)^2}\)
\(R=\left(1+\sqrt{a}+a\right).\frac{\left(1-\sqrt{a}\right)^2}{\left(1-\sqrt{a}\right)^2.\left(1+\sqrt{a}\right)^2}\)
\(=\left(1+\sqrt{a}\right)^2.\frac{1}{\left(1+\sqrt{a}\right)^2}=1\)
Bài 1: Rút gọn
a. \(\left(5-2\sqrt{3}\right)^2+\left(5+2\sqrt{3}\right)^2\)
b. \(\left(\sqrt{5}+\sqrt{2}\right)^2-\left(2\sqrt{5}+1\right)\left(2\sqrt{5}-1\right)-\sqrt{40}\)
c. \(\left(\sqrt{2}-1\right)^2-\frac{2}{3}\sqrt{4}+\frac{4\sqrt{2}}{5}+\sqrt{1\frac{11}{15}}-\sqrt{2}\)
d. \(\left(\sqrt{6}-\sqrt{18}+5\sqrt{2}-\frac{1}{2}\sqrt{8}\right)2\sqrt{6}+2\sqrt{3}\)
e. \(\left(2\sqrt{3}-3\sqrt{2}\right)^2+6\sqrt{6}+3\sqrt{24}\)
Bài 2: Rút gọn
A =\(\left(\frac{1}{x-\sqrt{x}}+\frac{1}{\sqrt{x}-1}:\frac{\sqrt{x+1}}{x-2\sqrt{x}+1}\right)\)(x>0 ; x khác 1)
Cho biểu thức: P = \(\left(1-\frac{\sqrt{x}}{\sqrt{x}+1}\right):\left(\frac{\sqrt{x}+3}{\sqrt{x}-2}+\frac{\sqrt{x}+2}{3-\sqrt{x}}+\frac{\sqrt{x}+2}{x-5\sqrt{x}+6}\right)\)
a, Rút gọn P
b, Tìm giá trị của x để P < 0