帮我出一些初中数学分数指数幂和平方根和实数相结合的计算培优题适合家教用数学式子表示

1. 化简 $\frac{2}{\sqrt{3}} \cdot \sqrt{27}$

解：$$\frac{2}{\sqrt{3}} \cdot \sqrt{27} = \frac{2 \cdot \sqrt{27}}{\sqrt{3}} = \frac{2 \cdot 3 \cdot \sqrt{3^3}}{\sqrt{3}} = 6 \sqrt{3^2} = 6 \cdot 3 = 18$$

答案：$18$

2. 计算 $\frac{4^3 \cdot 5^2}{2^2 \cdot 10^3}$

解：$$\frac{4^3 \cdot 5^2}{2^2 \cdot 10^3} = \frac{(2^2)^3 \cdot 5^2}{2^2 \cdot (2 \cdot 5)^3} = \frac{2^{2 \cdot 3} \cdot 5^2}{2^2 \cdot 2^3 \cdot 5^3} = \frac{2^4}{2 \cdot 5} = \frac{16}{10} = \frac{8}{5}$$

答案：$\frac{8}{5}$

3. 计算 $(\sqrt{3} + \sqrt{2})^2$

解：$$(\sqrt{3} + \sqrt{2})^2 = \sqrt{3}^2 + 2 \cdot \sqrt{3} \cdot \sqrt{2} + \sqrt{2}^2 = 3 + 2\sqrt{6} + 2 = 5 + 2\sqrt{6}$$

答案：$5 + 2\sqrt{6}$

4. 计算 $\sqrt{7 + \sqrt{48}}$

解：$$\sqrt{7 + \sqrt{48}} = \sqrt{4 + 3 + 2\sqrt{3^2 \cdot 2}} = \sqrt{(2 + 3) + 2\sqrt{3^2} \cdot \sqrt{2}} = \sqrt{2} + 3$$

答案：$\sqrt{2} + 3$

5. 化简 $\frac{x^3}{\sqrt{x^6}}$

解：$$\frac{x^3}{\sqrt{x^6}} = \frac{x^3}{(x^6)^{\frac{1}{2}}} = \frac{x^3}{x^3 \cdot x^3} = \frac{1}{x^3}$$

答案：$\frac{1}{x^3}$

6. 化简 $\frac{(a+b)^2 + (a-b)^2}{(a+b)^2 - (a-b)^2}$

解：$$\frac{(a+b)^2 + (a-b)^2}{(a+b)^2 - (a-b)^2} = \frac{a^2 + 2ab + b^2 + a^2 - 2ab + b^2}{a^2 + 2ab + b^2 - a^2 + 2ab - b^2} = \frac{2(a^2 + b^2)}{4ab} = \frac{a^2 + b^2}{2ab}$$

答案：$\frac{a^2 + b^2}{2ab}