Genişlet $$$\left(x^{2} + \frac{1}{\sqrt[3]{x}}\right)^{16}$$$

$$$\left(x^{2} + \frac{1}{\sqrt[3]{x}}\right)^{16}$$$ ifadesini genişletin.

Açılım aşağıdaki formülle verilir: $$$\left(a + b\right)^{n} = \sum_{k=0}^{n} {\binom{n}{k}} a^{n - k} b^{k}$$$, burada $$${\binom{n}{k}} = \frac{n!}{\left(n - k\right)! k!}$$$ ve $$$n! = 1 \cdot 2 \cdot \ldots \cdot n$$$.

Biliyoruz ki $$$a = x^{2}$$$, $$$b = \frac{1}{\sqrt[3]{x}}$$$ ve $$$n = 16$$$.

Dolayısıyla, $$$\left(x^{2} + \frac{1}{\sqrt[3]{x}}\right)^{16} = \sum_{k=0}^{16} {\binom{16}{k}} \left(x^{2}\right)^{16 - k} \left(\frac{1}{\sqrt[3]{x}}\right)^{k}.$$$

Şimdi, $$$0$$$ ile $$$16$$$ arasındaki her $$$k$$$ değeri için çarpımı hesaplayın.

$$$k = 0$$$: $$${\binom{16}{0}} \left(x^{2}\right)^{16 - 0} \left(\frac{1}{\sqrt[3]{x}}\right)^{0} = \frac{16!}{\left(16 - 0\right)! 0!} \left(x^{2}\right)^{16 - 0} \left(\frac{1}{\sqrt[3]{x}}\right)^{0} = x^{32}$$$

$$$k = 1$$$: $$${\binom{16}{1}} \left(x^{2}\right)^{16 - 1} \left(\frac{1}{\sqrt[3]{x}}\right)^{1} = \frac{16!}{\left(16 - 1\right)! 1!} \left(x^{2}\right)^{16 - 1} \left(\frac{1}{\sqrt[3]{x}}\right)^{1} = 16 x^{\frac{89}{3}}$$$

$$$k = 2$$$: $$${\binom{16}{2}} \left(x^{2}\right)^{16 - 2} \left(\frac{1}{\sqrt[3]{x}}\right)^{2} = \frac{16!}{\left(16 - 2\right)! 2!} \left(x^{2}\right)^{16 - 2} \left(\frac{1}{\sqrt[3]{x}}\right)^{2} = 120 x^{\frac{82}{3}}$$$

$$$k = 3$$$: $$${\binom{16}{3}} \left(x^{2}\right)^{16 - 3} \left(\frac{1}{\sqrt[3]{x}}\right)^{3} = \frac{16!}{\left(16 - 3\right)! 3!} \left(x^{2}\right)^{16 - 3} \left(\frac{1}{\sqrt[3]{x}}\right)^{3} = 560 x^{25}$$$

$$$k = 4$$$: $$${\binom{16}{4}} \left(x^{2}\right)^{16 - 4} \left(\frac{1}{\sqrt[3]{x}}\right)^{4} = \frac{16!}{\left(16 - 4\right)! 4!} \left(x^{2}\right)^{16 - 4} \left(\frac{1}{\sqrt[3]{x}}\right)^{4} = 1820 x^{\frac{68}{3}}$$$

$$$k = 5$$$: $$${\binom{16}{5}} \left(x^{2}\right)^{16 - 5} \left(\frac{1}{\sqrt[3]{x}}\right)^{5} = \frac{16!}{\left(16 - 5\right)! 5!} \left(x^{2}\right)^{16 - 5} \left(\frac{1}{\sqrt[3]{x}}\right)^{5} = 4368 x^{\frac{61}{3}}$$$

$$$k = 6$$$: $$${\binom{16}{6}} \left(x^{2}\right)^{16 - 6} \left(\frac{1}{\sqrt[3]{x}}\right)^{6} = \frac{16!}{\left(16 - 6\right)! 6!} \left(x^{2}\right)^{16 - 6} \left(\frac{1}{\sqrt[3]{x}}\right)^{6} = 8008 x^{18}$$$

$$$k = 7$$$: $$${\binom{16}{7}} \left(x^{2}\right)^{16 - 7} \left(\frac{1}{\sqrt[3]{x}}\right)^{7} = \frac{16!}{\left(16 - 7\right)! 7!} \left(x^{2}\right)^{16 - 7} \left(\frac{1}{\sqrt[3]{x}}\right)^{7} = 11440 x^{\frac{47}{3}}$$$

$$$k = 8$$$: $$${\binom{16}{8}} \left(x^{2}\right)^{16 - 8} \left(\frac{1}{\sqrt[3]{x}}\right)^{8} = \frac{16!}{\left(16 - 8\right)! 8!} \left(x^{2}\right)^{16 - 8} \left(\frac{1}{\sqrt[3]{x}}\right)^{8} = 12870 x^{\frac{40}{3}}$$$

$$$k = 9$$$: $$${\binom{16}{9}} \left(x^{2}\right)^{16 - 9} \left(\frac{1}{\sqrt[3]{x}}\right)^{9} = \frac{16!}{\left(16 - 9\right)! 9!} \left(x^{2}\right)^{16 - 9} \left(\frac{1}{\sqrt[3]{x}}\right)^{9} = 11440 x^{11}$$$

$$$k = 10$$$: $$${\binom{16}{10}} \left(x^{2}\right)^{16 - 10} \left(\frac{1}{\sqrt[3]{x}}\right)^{10} = \frac{16!}{\left(16 - 10\right)! 10!} \left(x^{2}\right)^{16 - 10} \left(\frac{1}{\sqrt[3]{x}}\right)^{10} = 8008 x^{\frac{26}{3}}$$$

$$$k = 11$$$: $$${\binom{16}{11}} \left(x^{2}\right)^{16 - 11} \left(\frac{1}{\sqrt[3]{x}}\right)^{11} = \frac{16!}{\left(16 - 11\right)! 11!} \left(x^{2}\right)^{16 - 11} \left(\frac{1}{\sqrt[3]{x}}\right)^{11} = 4368 x^{\frac{19}{3}}$$$

$$$k = 12$$$: $$${\binom{16}{12}} \left(x^{2}\right)^{16 - 12} \left(\frac{1}{\sqrt[3]{x}}\right)^{12} = \frac{16!}{\left(16 - 12\right)! 12!} \left(x^{2}\right)^{16 - 12} \left(\frac{1}{\sqrt[3]{x}}\right)^{12} = 1820 x^{4}$$$

$$$k = 13$$$: $$${\binom{16}{13}} \left(x^{2}\right)^{16 - 13} \left(\frac{1}{\sqrt[3]{x}}\right)^{13} = \frac{16!}{\left(16 - 13\right)! 13!} \left(x^{2}\right)^{16 - 13} \left(\frac{1}{\sqrt[3]{x}}\right)^{13} = 560 x^{\frac{5}{3}}$$$

$$$k = 14$$$: $$${\binom{16}{14}} \left(x^{2}\right)^{16 - 14} \left(\frac{1}{\sqrt[3]{x}}\right)^{14} = \frac{16!}{\left(16 - 14\right)! 14!} \left(x^{2}\right)^{16 - 14} \left(\frac{1}{\sqrt[3]{x}}\right)^{14} = \frac{120}{x^{\frac{2}{3}}}$$$

$$$k = 15$$$: $$${\binom{16}{15}} \left(x^{2}\right)^{16 - 15} \left(\frac{1}{\sqrt[3]{x}}\right)^{15} = \frac{16!}{\left(16 - 15\right)! 15!} \left(x^{2}\right)^{16 - 15} \left(\frac{1}{\sqrt[3]{x}}\right)^{15} = \frac{16}{x^{3}}$$$

$$$k = 16$$$: $$${\binom{16}{16}} \left(x^{2}\right)^{16 - 16} \left(\frac{1}{\sqrt[3]{x}}\right)^{16} = \frac{16!}{\left(16 - 16\right)! 16!} \left(x^{2}\right)^{16 - 16} \left(\frac{1}{\sqrt[3]{x}}\right)^{16} = \frac{1}{x^{\frac{16}{3}}}$$$

Dolayısıyla, $$$\left(x^{2} + \frac{1}{\sqrt[3]{x}}\right)^{16} = x^{32} + 16 x^{\frac{89}{3}} + 120 x^{\frac{82}{3}} + 560 x^{25} + 1820 x^{\frac{68}{3}} + 4368 x^{\frac{61}{3}} + 8008 x^{18} + 11440 x^{\frac{47}{3}} + 12870 x^{\frac{40}{3}} + 11440 x^{11} + 8008 x^{\frac{26}{3}} + 4368 x^{\frac{19}{3}} + 1820 x^{4} + 560 x^{\frac{5}{3}} + \frac{120}{x^{\frac{2}{3}}} + \frac{16}{x^{3}} + \frac{1}{x^{\frac{16}{3}}}.$$$

$$$\left(x^{2} + \frac{1}{\sqrt[3]{x}}\right)^{16} = 16 x^{\frac{89}{3}} + 120 x^{\frac{82}{3}} + 1820 x^{\frac{68}{3}} + 4368 x^{\frac{61}{3}} + 11440 x^{\frac{47}{3}} + 12870 x^{\frac{40}{3}} + 8008 x^{\frac{26}{3}} + 4368 x^{\frac{19}{3}} + 560 x^{\frac{5}{3}} + x^{32} + 560 x^{25} + 8008 x^{18} + 11440 x^{11} + 1820 x^{4} + \frac{16}{x^{3}} + \frac{120}{x^{\frac{2}{3}}} + \frac{1}{x^{\frac{16}{3}}}$$$A