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Popolare Algebra >

espandere (2a+b^2)^8

Soluzione

espandere

(2 a + b^{2})^{8}

Soluzione

256 a^{8} + 1024 a^{7} b^{2} + 1792 a^{6} b^{4} + 1792 a^{5} b^{6} + 1120 a^{4} b^{8} + 448 a^{3} b^{10} + 112 a^{2} b^{12} + 16 a b^{14} + b^{16}

Fasi della soluzione

(2 a + b^{2})^{8}

Applicare il teorema binomiale:

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

a = 2 a, b = b^{2}

= i = 0 \sum 8 (i 8) (2 a)^{(8 - i)} (b^{2})^{i}

Espandere la sommatoria

= \frac{8 !}{0 ! ( 8 - 0 ) !} (2 a)^{8} (b^{2})^{0} + \frac{8 !}{1 ! ( 8 - 1 ) !} (2 a)^{7} (b^{2})^{1} + \frac{8 !}{2 ! ( 8 - 2 ) !} (2 a)^{6} (b^{2})^{2} + \frac{8 !}{3 ! ( 8 - 3 ) !} (2 a)^{5} (b^{2})^{3} + \frac{8 !}{4 ! ( 8 - 4 ) !} (2 a)^{4} (b^{2})^{4} + \frac{8 !}{5 ! ( 8 - 5 ) !} (2 a)^{3} (b^{2})^{5} + \frac{8 !}{6 ! ( 8 - 6 ) !} (2 a)^{2} (b^{2})^{6} + \frac{8 !}{7 ! ( 8 - 7 ) !} (2 a)^{1} (b^{2})^{7} + \frac{8 !}{8 ! ( 8 - 8 ) !} (2 a)^{0} (b^{2})^{8}

Semplificare

\frac{8 !}{0 ! ( 8 - 0 ) !} (2 a)^{8} (b^{2})^{0} : 256 a^{8}

Semplificare

\frac{8 !}{1 ! ( 8 - 1 ) !} (2 a)^{7} (b^{2})^{1} : 1024 a^{7} b^{2}

Semplificare

\frac{8 !}{2 ! ( 8 - 2 ) !} (2 a)^{6} (b^{2})^{2} : 1792 a^{6} b^{4}

Semplificare

\frac{8 !}{3 ! ( 8 - 3 ) !} (2 a)^{5} (b^{2})^{3} : 1792 a^{5} b^{6}

Semplificare

\frac{8 !}{4 ! ( 8 - 4 ) !} (2 a)^{4} (b^{2})^{4} : 1120 a^{4} b^{8}

Semplificare

\frac{8 !}{5 ! ( 8 - 5 ) !} (2 a)^{3} (b^{2})^{5} : 448 a^{3} b^{10}

Semplificare

\frac{8 !}{6 ! ( 8 - 6 ) !} (2 a)^{2} (b^{2})^{6} : 112 a^{2} b^{12}

Semplificare

\frac{8 !}{7 ! ( 8 - 7 ) !} (2 a)^{1} (b^{2})^{7} : 16 a b^{14}

Semplificare

\frac{8 !}{8 ! ( 8 - 8 ) !} (2 a)^{0} (b^{2})^{8} : b^{16}

= 256 a^{8} + 1024 a^{7} b^{2} + 1792 a^{6} b^{4} + 1792 a^{5} b^{6} + 1120 a^{4} b^{8} + 448 a^{3} b^{10} + 112 a^{2} b^{12} + 16 a b^{14} + b^{16}

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