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H9
HT.Phong (9A5)
CTVHS
1 tháng 9 2023
a) \(A=\left(\dfrac{\sqrt{a}}{\sqrt{a}-1}-\dfrac{1}{a-\sqrt{a}}\right):\left(\dfrac{1}{\sqrt{a}+1}+\dfrac{2}{a-1}\right)\)
\(A=\left[\dfrac{\sqrt{a}}{\sqrt{a}-1}-\dfrac{1}{\sqrt{a}\left(\sqrt{a}-1\right)}\right]:\left(\dfrac{1}{\sqrt{a}+1}+\dfrac{2}{a-1}\right)\)
\(A=\left[\dfrac{a}{\sqrt{a}\left(\sqrt{a}-1\right)}-\dfrac{1}{\sqrt{a}\left(\sqrt{a}-1\right)}\right]:\left[\dfrac{1}{\sqrt{a}+1}+\dfrac{2}{a-1}\right]\)
\(A=\dfrac{a-1}{\sqrt{a}\left(\sqrt{a}-1\right)}:\left[\dfrac{\sqrt{a}-1}{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}+\dfrac{2}{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}\right]\)
\(A=\dfrac{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}{\sqrt{a}\left(\sqrt{a}-1\right)}:\dfrac{\sqrt{a}-1+2}{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}\)
\(A=\dfrac{\sqrt{a}+1}{\sqrt{a}}:\dfrac{\sqrt{a}+1}{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}\)
\(A=\dfrac{\sqrt{a}+1}{\sqrt{a}}\cdot\left(\sqrt{a}-1\right)\)
\(A=\dfrac{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}{\sqrt{a}}\)
\(A=\dfrac{a-1}{\sqrt{a}}\)
b) Ta có:
\(a=4+2\sqrt{3}=\left(\sqrt{3}\right)^2+2\sqrt{3}\cdot1+1^2=\left(\sqrt{3}+1\right)^2\)
Thay vào A ta có:
\(A=\dfrac{\left(\sqrt{3}+1\right)^2-1}{\sqrt{\left(\sqrt{3}+1\right)^2}}=\dfrac{4+2\sqrt{3}-1}{\sqrt{3}+1}=\dfrac{3+2\sqrt{3}}{\sqrt{3}+1}\)
c) \(A< 0\) khi:
\(\dfrac{a-1}{\sqrt{a}}< 0\)
Mà: \(\sqrt{a}\ge0\forall x\) (xác định)
\(\Leftrightarrow a-1< 0\)
\(\Leftrightarrow a< 1\)
Kết hợp với đk:
\(0< a< 1\)
26 tháng 9 2021
a) \(M=\dfrac{1+\sqrt{a}}{\sqrt{a}\left(\sqrt{a}-1\right)}.\dfrac{\left(\sqrt{a}-1\right)^2}{\sqrt{a}+1}=\dfrac{\sqrt{a}-1}{\sqrt{a}}\)
b) \(M=\dfrac{\sqrt{a}-1}{\sqrt{a}}=1-\dfrac{1}{\sqrt{a}}< 1\)
c) \(M=\dfrac{\sqrt{a}-1}{\sqrt{a}}=\dfrac{\sqrt{3-2\sqrt{2}}-1}{\sqrt{3-2\sqrt{2}}}=\dfrac{\sqrt{\left(\sqrt{2}-1\right)^2}-1}{\sqrt{\left(\sqrt{2}-1\right)^2}}=\dfrac{\sqrt{2}-1-1}{\sqrt{2}-1}=\dfrac{\sqrt{2}-2}{\sqrt{2}-1}\)
26 tháng 9 2021
\(a,M=\dfrac{1+\sqrt{a}}{\sqrt{a}\left(\sqrt{a}-1\right)}\cdot\dfrac{\left(\sqrt{a}-1\right)^2}{\sqrt{a}+1}=\dfrac{\sqrt{a}-1}{\sqrt{a}}\\ b,M=1-\dfrac{1}{\sqrt{a}}< 1\\ c,a=3-2\sqrt{2}\Leftrightarrow\sqrt{a}=\sqrt{\left(\sqrt{2}-1\right)^2}=\sqrt{2}-1\\ \Leftrightarrow M=\dfrac{\sqrt{2}-1-1}{\sqrt{2}-1}=\dfrac{\sqrt{2}-2}{\sqrt{2}-1}=\dfrac{-\sqrt{2}\left(\sqrt{2}-1\right)}{\sqrt{2}-1}=-\sqrt{2}\)
19 tháng 7 2018
a)\(3-\sqrt{3}+\sqrt{15}-3\sqrt{5}=\sqrt{3}\left(\sqrt{3}-1\right)-\sqrt{15}\left(\sqrt{3}-1\right)=\left(\sqrt{3}-1\right)\left(\sqrt{3}-\sqrt{15}\right)=\sqrt{3}\left(\sqrt{3}-1\right)\left(1-\sqrt{5}\right)\)\(\)b)\(\sqrt{1-a}+\sqrt{1-a^2}=\sqrt{1-a}.1+\sqrt{1-a}.\sqrt{1+a}=\sqrt{1-a}\left(\sqrt{1+a}+1\right)\)
19 tháng 7 2018
c)\(\sqrt{a^3}-\sqrt{b^3}+\sqrt{a^2b}-\sqrt{ab^2}=\left(\sqrt{a}-\sqrt{b}\right)\left(a+\sqrt{ab}+b\right)+\sqrt{ab}\left(\sqrt{a}-\sqrt{b}\right)=\left(\sqrt{a}-\sqrt{b}\right)\left(a+\sqrt{ab}+b+\sqrt{ab}\right)=\left(\sqrt{a}-\sqrt{b}\right)\left(a+2\sqrt{ab}+b\right)=\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{a}+\sqrt{b}\right)^2\)
21 tháng 2 2021
a) Ta có: \(B=\left(\dfrac{1}{a-\sqrt{a}}+\dfrac{1}{\sqrt{a}-1}\right):\dfrac{a-1}{a-2\sqrt{a}+1}\)
\(=\left(\dfrac{1}{\sqrt{a}\left(\sqrt{a}-1\right)}+\dfrac{\sqrt{a}}{\sqrt{a}\left(\sqrt{a}-1\right)}\right):\dfrac{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}{\left(\sqrt{a}-1\right)^2}\)
\(=\dfrac{\sqrt{a}+1}{\sqrt{a}\left(\sqrt{a}-1\right)}\cdot\dfrac{\sqrt{a}-1}{\sqrt{a}+1}\)
\(=\dfrac{1}{\sqrt{a}}\)
b) Thay \(a=3-2\sqrt{2}\) vào biểu thức \(B=\dfrac{1}{\sqrt{a}}\), ta được:
\(B=\dfrac{1}{\sqrt{3-2\sqrt{2}}}=\dfrac{1}{\sqrt{2}-1}=\sqrt{2}+1\)
Vậy: Khi \(a=3-2\sqrt{2}\) thì \(B=\sqrt{2}+1\)
AH
Akai Haruma
Giáo viên
23 tháng 8 2020
c)
$\sqrt{a^3}-\sqrt{b^3}+\sqrt{a^2b}-\sqrt{ab^2}$
$=(\sqrt{a^3}+\sqrt{a^2b})-(\sqrt{b^3}+\sqrt{ab^2})$
$=\sqrt{a^2}(\sqrt{a}+\sqrt{b})-\sqrt{b^2}(\sqrt{b}+\sqrt{a})$
$=a(\sqrt{a}+\sqrt{b})-b(\sqrt{b}+\sqrt{a})$
$=(\sqrt{a}+\sqrt{b})(a-b)=(\sqrt{a}+\sqrt{b})^2(\sqrt{a}-\sqrt{b})$
d)
$x-y+\sqrt{xy^2}-\sqrt{y^3}$
$=(x-y)+(\sqrt{xy^2}-\sqrt{y^3})$
$=(\sqrt{x}-\sqrt{y})(\sqrt{x}+\sqrt{y})+y(\sqrt{x}-\sqrt{y})$
$=(\sqrt{x}-\sqrt{y})(\sqrt{x}+\sqrt{y}+y)$
AH
Akai Haruma
Giáo viên
23 tháng 8 2020
a)
$3-\sqrt{3}+\sqrt{15}-3\sqrt{15}$
$=\sqrt{3}(\sqrt{3}-1)-\sqrt{15}(3-1)$
$=(\sqrt{3}(\sqrt{3}-1)-\sqrt{15}(\sqrt{3}+1)(\sqrt{3}-1)$
$=(\sqrt{3}-1)[\sqrt{3}-\sqrt{15}(\sqrt{3}+1)]$
$=(\sqrt{3}-1)(\sqrt{3}-\sqrt{45}-\sqrt{15})$
b)
$\sqrt{1-a}+\sqrt{1-a^2}=\sqrt{1-a}+\sqrt{(1-a)(1+a)}$
$=\sqrt{1-a}+\sqrt{1-a}.\sqrt{1+a}=\sqrt{1-a}(1+\sqrt{1+a})$
NV
Nguyễn Việt Lâm
Giáo viên
21 tháng 8 2020
a/
Bạn coi lại đề, số cuối là \(3\sqrt{15}\) hay \(3\sqrt{5}\)
b/
\(=\sqrt{1-a}+\sqrt{\left(1-a\right)\left(1+a\right)}=\sqrt{1-a}\left(1+\sqrt{1+a}\right)\)
c/
\(=\sqrt{a^3}+\sqrt{a^2b}-\sqrt{b^3}-\sqrt{ab^2}\)
\(=\sqrt{a^2}\left(\sqrt{a}+\sqrt{b}\right)-\sqrt{b^2}\left(\sqrt{a}+\sqrt{b}\right)\)
\(=\left(a-b\right)\left(\sqrt{a}+\sqrt{b}\right)\)
(Hoặc có thể biến đổi thêm \(=\left(\sqrt{a}+\sqrt{b}\right)^2\left(\sqrt{a}-\sqrt{b}\right)\) cũng được)
d/
\(=\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)+\sqrt{y^2}\left(\sqrt{x}-\sqrt{y}\right)\)
\(=\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}+y\right)\)