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(\(\sqrt{x^2-6x+13}\) - \(\sqrt{x^2-6x+10}\))(\(\sqrt{x^2-6x+13}\) + \(\sqrt{x^2-6x+10}\)) = x2 - 6x + 13 - x2 + 6x - 10 = 3
=>
\(\sqrt{x^2-6x+13}\) + \(\sqrt{x^2-6x+10}\) = 3
Ta có :
\(\sqrt{x^2-6x+13}-\sqrt{x^2-6x+10}\)
=\(\sqrt{x^2-2.3.x+3^2+4}-\sqrt{x^2-2.3.x+3^2+1}\)
=\(\sqrt{\left(x-3\right)^2+2^2}-\sqrt{\left(x-3\right)^2+1^2}\)
Ta có :
\(\sqrt{x^2-6x+13}+\sqrt{x^2-6x+10}\)
\(=\sqrt{x^2-6x+9+4}+\sqrt{x^2-6x+9+1}\)
\(=\sqrt{\left(x-3\right)^2+2^2}+\sqrt{\left(x-3\right)^2+1}\)
Lời giải:
Bạn cứ nhớ công thức $\sqrt{x^2}=|x|$, rồi dùng điều kiện đề bài để phá dấu trị tuyệt đối là được
a)
\(\sqrt{16a^2}-5a=\sqrt{(4a)^2}-5a=|4a|-5a=4a-5a=-a\)
b)
\(3x+2-\sqrt{9x^2+6x+1}=3x+2-\sqrt{(3x)^2+2.3x.1+1^2}\)
\(=3x+2-\sqrt{(3x+1)^2}=3x+2-|3x+1|=3x+2-(3x+1)=1\)
c)
\(\sqrt{8+2\sqrt{7}}-\sqrt{7}=\sqrt{7+1+2.\sqrt{7}.\sqrt{1}}-\sqrt{7}\)
\(=\sqrt{(\sqrt{7}+1)^2}-\sqrt{7}=|\sqrt{7}+1|-\sqrt{7}=\sqrt{7}+1-\sqrt{7}=1\)
d)
\(\sqrt{14-2\sqrt{13}}+\sqrt{14+2\sqrt{13}}=\sqrt{13+1-2\sqrt{13}}+\sqrt{13+1+2\sqrt{13}}\)
\(=\sqrt{(\sqrt{13}-1)^2}+\sqrt{(\sqrt{13}+1)^2}=|\sqrt{13}-1|+|\sqrt{13}+1|\)
\(=\sqrt{13}-1+\sqrt{13}+1=2\sqrt{13}\)
e)
\(2x-\sqrt{4x^2-4x+1}=2x-\sqrt{(2x-1)^2}=2x-|2x-1|=2x-(2x-1)=1\)
g)
\(|x-2|+\frac{\sqrt{x^2-4x+4}}{x-2}=|x-2|+\frac{\sqrt{(x-2)^2}}{x-2}=|x-2|+\frac{|x-2|}{x-2}\)
\(=(x-2)+\frac{(x-2)}{x-2}=x-2+1=x-1\)
a)\(\frac{3}{2}\sqrt{6}+2\sqrt{\frac{2}{3}}-4\sqrt{\frac{3}{2}}=\frac{3}{2}\sqrt{6}+2\frac{\sqrt{6}}{3}-4\frac{\sqrt{6}}{2}\)
\(=\sqrt{6}\left(\frac{3}{2}+\frac{2}{3}-\frac{4}{2}\right)=\sqrt{6}.\frac{1}{6}\)
b) \(\left(x\sqrt{\frac{6}{x}}+\sqrt{\frac{2x}{3}}+\sqrt{6x}\right):\sqrt{6x}=\left(x.\frac{\sqrt{6x}}{x}+\frac{\sqrt{6x}}{3}+\sqrt{6x}\right):\sqrt{6x}\)
\(=1+\frac{1}{3}+1=2\frac{1}{3}\)
2) Dễ thấy\(\left(\sqrt{x^2-6x+13}-\sqrt{x^2-6x+10}\right)\left(\sqrt{x^2-6x+13}+\sqrt{x^2-6x+10}\right)=x^2-6x+13-x^2+6x-10=3\)
\(\Leftrightarrow1.\left(\sqrt{x^2-6x+13}+\sqrt{x^2-6x+10}\right)=3\)
\(\Leftrightarrow\sqrt{x^2-6x+13}+\sqrt{x^2-6x+10}=3\)
Ta có: a+ b= \(\frac{-1+\sqrt{2}}{2}\) + \(\frac{-1-\sqrt{2}}{2}\)= -1
a*b = \(\frac{-1+\sqrt{2}}{2}\)* \(\frac{-1-\sqrt{2}}{2}\)= -\(\frac{1}{4}\)
a2 + b2 = (a+ b)2 - 2ab = 1+ \(\frac{1}{2}\)= \(\frac{3}{2}\)
a4 + b4 = (a2 + b2 )2 - 2a2b2 = \(\frac{9}{4}\)- \(\frac{1}{8}\)= \(\frac{17}{8}\)
a3 + b3 = ( a + b)3 - 3ab(a + b ) = -1-\(\frac{3}{4}\)= \(\frac{-7}{4}\)
vay a7 + b7 = (a3 + b3 )(a4 + b4 ) -a3b3(a+b)= \(\frac{-7}{4}\)* \(\frac{17}{8}\)- (-\(\frac{1}{64}\)) * (-1) = \(\frac{-239}{64}\)