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5.已知a是$\sqrt {7}$的整数部分,b是$\sqrt {7}$的小数部分,求$a-2b$的值.
答案
$\because 2 ＜ \sqrt{7} ＜ 3$，$\therefore a = 2$，$b = \sqrt{7} - 2$，$\therefore a - 2b = 2 - 2(\sqrt{7} - 2) = 6 - 2\sqrt{7}$．
6.计算:
(1)$3\sqrt {2}-\frac {1}{2}\sqrt {3}+\frac {1}{3}\sqrt {2}-\sqrt {2}+\sqrt {3}$; (2)$\sqrt [3]{-27}+|3-\sqrt {5}|-(\sqrt {9}-\sqrt [3]{8})^{2}+3\sqrt {5}$;
(3)$\sqrt {4}-\sqrt [3]{8}-\sqrt [3]{-\frac {1}{27}}-(-\frac {1}{3})^{2}$.
(1)$3\sqrt {2}-\frac {1}{2}\sqrt {3}+\frac {1}{3}\sqrt {2}-\sqrt {2}+\sqrt {3}$; (2)$\sqrt [3]{-27}+|3-\sqrt {5}|-(\sqrt {9}-\sqrt [3]{8})^{2}+3\sqrt {5}$;
(3)$\sqrt {4}-\sqrt [3]{8}-\sqrt [3]{-\frac {1}{27}}-(-\frac {1}{3})^{2}$.
答案
(1) $\frac{7}{3}\sqrt{2} + \frac{1}{2}\sqrt{3}$
(2) $2\sqrt{5} - 1$
(3) $\frac{2}{9}$
(2) $2\sqrt{5} - 1$
(3) $\frac{2}{9}$
7.如图1,这是由8个同样大小的正方体组成的魔方,总体积为$64cm^
{3}$.
(1)这个魔方的棱长为____;
(2)图1中阴影部分是一个正方形ABCD,求这个正方形的边长;
(3)把正方形ABCD放置在数轴上,如图2所示,使得点A与数1表示的点重合,则点D在数轴上表示的数为____.
{3}$.(1)这个魔方的棱长为____;
(2)图1中阴影部分是一个正方形ABCD,求这个正方形的边长;
(3)把正方形ABCD放置在数轴上,如图2所示,使得点A与数1表示的点重合,则点D在数轴上表示的数为____.
答案
(1) $4\mathrm{cm}$
(2) 设小正方体的棱长为$x\mathrm{cm}$，根据题意，得$8x^3 = 64$，$\therefore x = 2$，$\therefore S_{\text{阴影正方形}} = 4 \times \frac{1}{2} \times 2 \times 2 = 8(\mathrm{cm}^2)$，$\therefore CD = 2\sqrt{2}\mathrm{cm}$，答：这个正方形的边长是$2\sqrt{2}\mathrm{cm}$．
(3) $1 - 2\sqrt{2}$
(2) 设小正方体的棱长为$x\mathrm{cm}$，根据题意，得$8x^3 = 64$，$\therefore x = 2$，$\therefore S_{\text{阴影正方形}} = 4 \times \frac{1}{2} \times 2 \times 2 = 8(\mathrm{cm}^2)$，$\therefore CD = 2\sqrt{2}\mathrm{cm}$，答：这个正方形的边长是$2\sqrt{2}\mathrm{cm}$．
(3) $1 - 2\sqrt{2}$
