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c: Ta có: \(\sqrt{\left(4+\sqrt{10}\right)^2}-\sqrt{\left(4-\sqrt{10}\right)^2}\)
\(=4+\sqrt{10}-4+\sqrt{10}\)
\(=2\sqrt{10}\)
d: Ta có: \(\sqrt{3-2\sqrt{2}}+\sqrt{6-4\sqrt{2}}+\sqrt{9-4\sqrt{2}}\)
\(=\sqrt{2}-1+2-\sqrt{2}+2\sqrt{2}-1\)
\(=2\sqrt{2}\)
a) \(=\left(2\sqrt{3}\right)^2-\left(3\sqrt{2}\right)^2=12-18=-6\)
b) \(=\dfrac{\sqrt{2013}+\sqrt{2014}}{2013-2014}-\dfrac{\sqrt{2014}+\sqrt{2015}}{2014-2015}=-\sqrt{2013}-\sqrt{2014}+\sqrt{2014}-\sqrt{2015}=-\sqrt{2013}-\sqrt{2015}\)
c) \(=4+\sqrt{10}-4+\sqrt{10}=2\sqrt{10}\)
d) \(=\sqrt{\left(\sqrt{2}-1\right)^2}+\sqrt{\left(2-\sqrt{2}\right)^2}+\sqrt{\left(2\sqrt{2}-1\right)^2}=\sqrt{2}-1+2-\sqrt{2}+2\sqrt{2}-1=2\sqrt{2}\)
Ta có: \(\left(x+\sqrt{x^2+2013}\right)\left(y+\sqrt{y^2+2013}\right)=2013\)
\(\Leftrightarrow\left(x-\sqrt{x^2+2013}\right)\left(x+\sqrt{x^2+2013}\right)\left(y+\sqrt{y^2+2013}\right)=2013\left(x-\sqrt{x^2+2013}\right)\)
\(\Leftrightarrow-2013\left(y+\sqrt{y^2+2013}\right)=2013\left(x-\sqrt{x^2+2013}\right)\)
\(\Leftrightarrow-y-\sqrt{y^2+2013}=x-\sqrt{x^2+2013}\)
⇔\(x+y=\sqrt{x^2+2013}-\sqrt{y^2+2013}\)(1)
Nhân liên hợp tương tự nhân \(y-\sqrt{y^2+2013}\)vào hai về rút được
\(x+y=\sqrt{y^2+2013}-\sqrt{x^2+2013}\)(2)
Cộng vế theo vế (1)(2) ta được \(x+y=0\Rightarrow x=-y\)
Thay vào \(A=\left(-y\right)^{2014}-y^{2014}+1=1\)
\(\sqrt{\left(x-2013\right)^{10}}+\sqrt{\left(x-2014\right)^{14}}=1\)
Mà \(\sqrt{\left(x-2013\right)^{10}};\sqrt{\left(x-2014\right)^{14}}\ge0\)
=> \(\sqrt{\left(x-2013\right)^{10}}=0;\sqrt{\left(x-2014\right)^{14}}=1\)
HOặc \(\sqrt{\left(x-2013\right)^{10}}=1;\sqrt{\left(x-2013\right)^{14}}=0\)
\(\sqrt{\left(x-2013\right)^{10}}=1\rightarrow x-2013\in\left\{-1;1\right\};x\in\left\{2014;2012\right\}\)
\(\sqrt{\left(x-2014\right)^{14}}=0;x-2014=0;x=2014\)
=> x = 2014 (thích hợp)
\(\sqrt{\left(x-2013\right)^{10}}=0;x-2013=0;x=2013\)
\(\sqrt{\left(x-2014\right)^{14}}=1;x-2014\in\left\{-1;1\right\};x\in\left\{2013;2015\right\}\)
=> x = 2013 (thích hợp)
Vậy x = 2013 hoặc x = 2014
Đặt \(x-2003=t\)
Ta có: \(\sqrt{t^{10}}+\sqrt{\left(1-t\right)^{14}}=1\Leftrightarrow\left|t\right|^5+\left|1-t\right|^7=1\text{(*)}\)
\(\left(\text{*}\right)\Rightarrow\left|t\right|;\left|1-t\right|\le1\)
\(+t1\text{ (loại)}\)
\(+t=0\) thì \(\left(\text{*}\right)\) thỏa
\(+0
Ta có:
\(\left(x+\sqrt{x^2+2013}\right)\left(y+\sqrt{y^2+2013}\right)=2013\\ \Leftrightarrow\left(x^2-x^2-2013\right)\left(y+\sqrt{y^2+2013}\right)=2013\left(x-\sqrt{x^2+2013}\right)\\ \Leftrightarrow y+\sqrt{y^2+2013}=\sqrt{x^2+2013}-x\left(1\right)\)
Tương tự: \(x+\sqrt{x^2+2013}=\sqrt{y^2+2013}-y\left(2\right)\)
Do đó: 2x=-2y
Suy ra: x=-y
Do đó:
\(x^{2013}+y^{2013}=\left(-y\right)^{2013}+y^{2013}=0\left(ĐPCM\right)\)
\(\left(x+\sqrt{x^2+\sqrt{2013}}\right)\left(x-\sqrt{x^2+\sqrt{2013}}\right)=x^2-x^2-\sqrt{2013}=-\sqrt{2013}\) (1)
Theo đề bài và (1) => dpcm
b) theo a có \(y+\sqrt{y^2+\sqrt{2013}}=-x+\sqrt{x^2+\sqrt{2013}}\)(2)
tương tự ta có \(x+\sqrt{x^2+\sqrt{2013}}=-y+\sqrt{y^2+\sqrt{2013}}\)(3)
Cộng 2 vế (2) với (3) => x+y = -x -y
hay 2(x+y) =0 =>S= x+y =0
Có ab > 2013a + 2014b <=> 1 > 2013/b + 2014/a (vì a,b >0 )
\(\Leftrightarrow a+b>\frac{2013\left(a+b\right)}{b}+\frac{2014\left(a+b\right)}{a}=2013+2014+\frac{2013a}{b}+\frac{2014b}{a}\)
Mà \(\frac{2013a}{b}+\frac{2014b}{a}\ge2\sqrt{2013\cdot2014}\)
\(\Rightarrow a+b>2013+2014+2\sqrt{2013\cdot2014}=\left(\sqrt{2013}+\sqrt{2014}\right)^2\)
=> đpcm
Tích cho mk nhoa !!!! ~~~
\(\left(\sqrt{2014}-\sqrt{2013}\right).\left(\sqrt{2014}+\sqrt{2013}\right)\)
=> \(\sqrt{2014^2}-\sqrt{2013^2}\)
=> \(2014-2013\)
\(=1\)
Vậy ..............
\(\left(\sqrt{2014}-\sqrt{2013}\right).\left(\sqrt{2014}+\sqrt{2013}\right)=1\)
\(VT=\left(\sqrt{2014}-\sqrt{2013}\right).\left(\sqrt{2014}+\sqrt{2013}\right)\)
\(=\sqrt{2014^2}-\sqrt{2013^2}\)
\(=2014-2013\)
\(=1=VP\left(dpcm\right)\)
Vậy....