项目式学习【问题情境】数学活动课上，老师带领同学们开展“运用规律求一个正数的算术平方根和立方根”的实践活动，同学们列出了表1中的算术平方根和表2中的立方根如下：
表1：
| $x$ | $···$ | $0.006\ 4$ | $0.64$ | $64$ | $6\ 400$ | $640\ 000$ | $···$ |
| --- | --- | --- | --- | --- | --- | --- | --- |
| $\sqrt{x}$ | $···$ | $0.08$ | $0.8$ | | $80$ | $800$ | $···$ |
表2：
| $x$ | $···$ | $0.000\ 064$ | $0.064$ | $64$ | $64\ 000$ | $64\ 000\ 000$ | $···$ |
| --- | --- | --- | --- | --- | --- | --- | --- |
| $\sqrt[3]{x}$ | $···$ | $0.04$ | $0.4$ | | $40$ | $400$ | $···$ |
【探索发现】(1)根据上述探究,可以得到被开方数和它的算术平方根和立方根之间小数点的变化规律是:若被开方数的小数点向右或向左移动位,则它的算术平方根的小数点就相应地向右或向左移动位;若被开方数的小数点向右或向左移动位,则它的立方根的小数点就相应地向右或向左移动位.
【规律应用】(2)请运用上述规律,解答下列问题:
①已知$\sqrt{3}\approx 1.732,\sqrt[3]{3}\approx 1.442$,则$\sqrt{300}\approx$,$\sqrt[3]{0.003}\approx$;
②若$\sqrt{a}\approx 14.142,\sqrt[3]{0.7}\approx b$,求$a,b$的值.
(参考数据:$\sqrt{2}\approx 1.414\ 2,\sqrt{20}\approx 4.472\ 1,\sqrt[3]{7}\approx 1.912\ 9,\sqrt[3]{700}\approx 8.879$)
(3)运用上述规律,你能根据$\sqrt{3}$的值求出$\sqrt{30}$的值吗?请说明理由.
(4)已知$9.97^2=99.400\ 9,9.98^2=99.600\ 4,9.99^2=99.800\ 1$,求$\sqrt{997\ 000}$的个位数字.

(1)2 1 3 1
(2)①17.32 0.144 2
②
∵$\sqrt{2}\approx 1.414 2,\sqrt[3]{700}\approx 8.879,\therefore \sqrt{200}\approx 14.142,\sqrt[3]{0.7}\approx 0.887 9,\therefore a=200,b=0.887 9.$
(3)不能.理由:$\because \sqrt{3}\approx 1.732$,运用上述规律可得$\sqrt{0.03}\approx 0.173 2,\sqrt{300}\approx 17.32,\therefore$不能求出$\sqrt{30}$的值.
(4)$\because 9.97^2 = 99.400 9,9.98^2 = 99.600 4,9.99^2 = 99.800 1$,
又$\because \sqrt{99.600 4}＜\sqrt{99.7}＜\sqrt{99.800 1},\therefore 9.98＜\sqrt{99.7}＜9.99$,
$\therefore 998＜\sqrt{997 000}＜999$,即其个位数字为 8.