Answer:
[tex] \\ 64^{\frac{3}{4}} = 16\sqrt{2} [/tex]
Step-by-step explanation:
We need here to remember that:
[tex] \\{(x^{a})}^{b} = x^{a * b} [/tex]
[tex] \\{x^{a} * x^{b} = x^{a + b} [/tex]
Then,
[tex] \\ 64^{\frac{3}{4}} = {(8^{2})}^{(\frac{3}{4})} = {{(2^{3})}^{2}}^\frac{3}{4} [/tex]
[tex] \\ 64^{\frac{3}{4}} = {{2^{3}}^2}^\frac{3}{4} = 2^\frac{3*2*3}{4} [/tex]
[tex] \\ 64^{\frac{3}{4}} = 2^\frac{2*3*3}{4} = 2^\frac{3*3}{2} = 2^\frac{9}{2} [/tex]
[tex] \\ 64^{\frac{3}{4}} = 2^{\frac{9}{2}} [/tex]
Since [tex] \\ \frac{9}{2} = \frac{4}{2} + \frac{4}{2} + \frac{1}{2} [/tex]
[tex] \\ 64^{\frac{3}{4}} = 2^{\frac{9}{2}} = {2^{(\frac{4}{2} + \frac{4}{2} + \frac{1}{2})} [/tex]
[tex] \\ 64^{\frac{3}{4}} = 2^{\frac{4}{2}} * 2^{\frac{4}{2}} * {2}^{\frac{1}{2} [/tex]
[tex] \\ 64^{\frac{3}{4}} = 2^{2} * 2^{2} * {2}^{\frac{1}{2} [/tex]
[tex] \\ 64^{\frac{3}{4}} = 4 * 4 * \sqrt{2} = 16 \sqrt{2} [/tex]
