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Popolare Algebra >

espandere (1+sqrt(5))^{14}

Soluzione

espandere

(1 + 5)^{14}

Soluzione

6905856 + 30883845

+1

Decimale

13811692.56462 \dots

Fasi della soluzione

(1 + 5)^{14}

Applicare il teorema binomiale:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = 1, b = 5

= i = 0 \sum 14 (i 14) \cdot 1^{(14 - i)} (5)^{i}

Espandere la sommatoria

= \frac{14 !}{0 ! ( 14 - 0 ) !} \cdot 1^{14} (5)^{0} + \frac{14 !}{1 ! ( 14 - 1 ) !} \cdot 1^{13} (5)^{1} + \frac{14 !}{2 ! ( 14 - 2 ) !} \cdot 1^{12} (5)^{2} + \frac{14 !}{3 ! ( 14 - 3 ) !} \cdot 1^{11} (5)^{3} + \frac{14 !}{4 ! ( 14 - 4 ) !} \cdot 1^{10} (5)^{4} + \frac{14 !}{5 ! ( 14 - 5 ) !} \cdot 1^{9} (5)^{5} + \frac{14 !}{6 ! ( 14 - 6 ) !} \cdot 1^{8} (5)^{6} + \frac{14 !}{7 ! ( 14 - 7 ) !} \cdot 1^{7} (5)^{7} + \frac{14 !}{8 ! ( 14 - 8 ) !} \cdot 1^{6} (5)^{8} + \frac{14 !}{9 ! ( 14 - 9 ) !} \cdot 1^{5} (5)^{9} + \frac{14 !}{10 ! ( 14 - 10 ) !} \cdot 1^{4} (5)^{10} + \frac{14 !}{11 ! ( 14 - 11 ) !} \cdot 1^{3} (5)^{11} + \frac{14 !}{12 ! ( 14 - 12 ) !} \cdot 1^{2} (5)^{12} + \frac{14 !}{13 ! ( 14 - 13 ) !} \cdot 1^{1} (5)^{13} + \frac{14 !}{14 ! ( 14 - 14 ) !} \cdot 1^{0} (5)^{14}

Semplificare

\frac{14 !}{0 ! ( 14 - 0 ) !} \cdot 1^{14} (5)^{0} : 1

Semplificare

\frac{14 !}{1 ! ( 14 - 1 ) !} \cdot 1^{13} (5)^{1} : 145

Semplificare

\frac{14 !}{2 ! ( 14 - 2 ) !} \cdot 1^{12} (5)^{2} : 455

Semplificare

\frac{14 !}{3 ! ( 14 - 3 ) !} \cdot 1^{11} (5)^{3} : 18205

Semplificare

\frac{14 !}{4 ! ( 14 - 4 ) !} \cdot 1^{10} (5)^{4} : 25025

Semplificare

\frac{14 !}{5 ! ( 14 - 5 ) !} \cdot 1^{9} (5)^{5} : 500505

Semplificare

\frac{14 !}{6 ! ( 14 - 6 ) !} \cdot 1^{8} (5)^{6} : 375375

Semplificare

\frac{14 !}{7 ! ( 14 - 7 ) !} \cdot 1^{7} (5)^{7} : 4290005

Semplificare

\frac{14 !}{8 ! ( 14 - 8 ) !} \cdot 1^{6} (5)^{8} : 1876875

Semplificare

\frac{14 !}{9 ! ( 14 - 9 ) !} \cdot 1^{5} (5)^{9} : 12512505

Semplificare

\frac{14 !}{10 ! ( 14 - 10 ) !} \cdot 1^{4} (5)^{10} : 3128125

Semplificare

\frac{14 !}{11 ! ( 14 - 11 ) !} \cdot 1^{3} (5)^{11} : 11375005

Semplificare

\frac{14 !}{12 ! ( 14 - 12 ) !} \cdot 1^{2} (5)^{12} : 1421875

Semplificare

\frac{14 !}{13 ! ( 14 - 13 ) !} \cdot 1^{1} (5)^{13} : 2187505

Semplificare

\frac{14 !}{14 ! ( 14 - 14 ) !} \cdot 1^{0} (5)^{14} : 78125

= 1 + 145 + 455 + 18205 + 25025 + 500505 + 375375 + 4290005 + 1876875 + 12512505 + 3128125 + 11375005 + 1421875 + 2187505 + 78125

1 + 145 + 455 + 18205 + 25025 + 500505 + 375375 + 4290005 + 1876875 + 12512505 + 3128125 + 11375005 + 1421875 + 2187505 + 78125 = 6905856 + 30883845

= 6905856 + 30883845

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