数学

每日一题（第二十六天）

$1.已知a＝\dfrac{1}{2}\sqrt{\sqrt{2}+\dfrac{1}{8}}-\dfrac{\sqrt{2}}{8}，$

$求代数式a^2+\sqrt{a^4+a+1}的值.$

$2.已知非零实数a，b，c满足\dfrac{1}{a}+\dfrac{2}{b}+\dfrac{3}{c}＝0，$

$求代数式\dfrac{3b+2c}{6a}+\dfrac{2c+6a}{3b}+\dfrac{6a+3b}{2c}的值.$

$3.已知实数a，b，c满足（a+b+c）（\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}）＝1，$

$求证：\dfrac{1}{（a+b+c）^{2027}}＝\dfrac{1}{a^{2027}}+\dfrac{1}{b^{2027}}+\dfrac{1}{c^{2027}}.$

$4.实数a，b，x，y满足：$

$ax+by＝3$

$ax^2+by^2＝7$

$ax^3+by^3＝16$

$ax^4+by^4＝42$

$求ax^5+by^5的值.$

$5.已知非零实数a，b，c满足a+b+c＝0，求：$

$（1）（\dfrac{a-b}{c}+\dfrac{b-c}{a}+\dfrac{c-a}{b}）（\dfrac{c}{a-b}+\dfrac{a}{b-c}+\dfrac{b}{c-a}）；$

$（2）\dfrac{a^5+b^5+c^5}{abc（ab+bc+ca）}.$

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