1
Simplify these expressions (express with positive indices):
a) $$ x^{\frac{2}{3}} \times x^{\frac{4}{3}} $$
b) $$ \frac{a^5}{a^{-2}} $$
c) $$ (b^4)^{\frac{3}{2}} $$
d) $$ 3x^{\frac{1}{2}} \div 6x^{-\frac{1}{2}} $$
e) $$ \sqrt[4]{81y^8} $$
答案：
a) $$ x^{\frac{2}{3} + \frac{4}{3}} = x^2 $$
b) $$ a^{5 - (-2)} = a^7 $$
c) $$ b^{4 \times \frac{3}{2}} = b^6 $$
d) $$ \frac{3}{6} x^{\frac{1}{2} - (-\frac{1}{2})} = \frac{1}{2} x^1 = \frac{1}{2}x $$
e) $$ (81y^8)^{\frac{1}{4}} = 81^{\frac{1}{4}} y^{8 \times \frac{1}{4}} = 3 y^2 $$
2
Calculate these values:
a) $$ 16^{\frac{1}{2}} $$
b) $$ 27^{\frac{1}{3}} $$
c) $$ 16^{\frac{3}{4}} $$
d) $$ 9^{-\frac{1}{2}} $$
答案：
a) $$ 16^{\frac{1}{2}} = \sqrt{16} = 4 $$
b) $$ 27^{\frac{1}{3}} = \sqrt[3]{27} = 3 $$
c) $$ 16^{\frac{3}{4}} = (\sqrt[4]{16})^3 = 2^3 = 8 $$
d) $$ 9^{-\frac{1}{2}} = \frac{1}{9^{\frac{1}{2}}} = \frac{1}{\sqrt{9}} = \frac{1}{3} $$