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Popolare Algebra >

espandere (\sqrt[3]{27x}+2y)^7

Soluzione

espandere

(3 27 x + 2 y)^{7}

Soluzione

2187 3 x^{7} + 10206 x^{2} y + 20412 y^{2} 3 x^{5} + 22680 y^{3} 3 x^{4} + 15120 x y^{4} + 6048 y^{5} (3 x)^{2} + 1344 y^{6} 3 x + 128 y^{7}

Fasi della soluzione

(3 27 x + 2 y)^{7}

Applicare il teorema binomiale:

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

a = 3 27 x, b = 2 y

= i = 0 \sum 7 (i 7) (3 27 x)^{(7 - i)} (2 y)^{i}

Espandere la sommatoria

= \frac{7 !}{0 ! ( 7 - 0 ) !} (3 27 x)^{7} (2 y)^{0} + \frac{7 !}{1 ! ( 7 - 1 ) !} (3 27 x)^{6} (2 y)^{1} + \frac{7 !}{2 ! ( 7 - 2 ) !} (3 27 x)^{5} (2 y)^{2} + \frac{7 !}{3 ! ( 7 - 3 ) !} (3 27 x)^{4} (2 y)^{3} + \frac{7 !}{4 ! ( 7 - 4 ) !} (3 27 x)^{3} (2 y)^{4} + \frac{7 !}{5 ! ( 7 - 5 ) !} (3 27 x)^{2} (2 y)^{5} + \frac{7 !}{6 ! ( 7 - 6 ) !} (3 27 x)^{1} (2 y)^{6} + \frac{7 !}{7 ! ( 7 - 7 ) !} (3 27 x)^{0} (2 y)^{7}

Semplificare

\frac{7 !}{0 ! ( 7 - 0 ) !} (3 27 x)^{7} (2 y)^{0} : 2187 (3 x)^{7}

Semplificare

\frac{7 !}{1 ! ( 7 - 1 ) !} (3 27 x)^{6} (2 y)^{1} : 10206 x^{2} y

Semplificare

\frac{7 !}{2 ! ( 7 - 2 ) !} (3 27 x)^{5} (2 y)^{2} : 20412 y^{2} (3 x)^{5}

Semplificare

\frac{7 !}{3 ! ( 7 - 3 ) !} (3 27 x)^{4} (2 y)^{3} : 22680 y^{3} (3 x)^{4}

Semplificare

\frac{7 !}{4 ! ( 7 - 4 ) !} (3 27 x)^{3} (2 y)^{4} : 15120 x y^{4}

Semplificare

\frac{7 !}{5 ! ( 7 - 5 ) !} (3 27 x)^{2} (2 y)^{5} : 6048 y^{5} (3 x)^{2}

Semplificare

\frac{7 !}{6 ! ( 7 - 6 ) !} (3 27 x)^{1} (2 y)^{6} : 1344 y^{6} 3 x

Semplificare

\frac{7 !}{7 ! ( 7 - 7 ) !} (3 27 x)^{0} (2 y)^{7} : 128 y^{7}

= 2187 (3 x)^{7} + 10206 x^{2} y + 20412 y^{2} (3 x)^{5} + 22680 y^{3} (3 x)^{4} + 15120 x y^{4} + 6048 y^{5} (3 x)^{2} + 1344 y^{6} 3 x + 128 y^{7}

Semplificare

2187 (3 x)^{7} + 10206 x^{2} y + 20412 y^{2} (3 x)^{5} + 22680 y^{3} (3 x)^{4} + 15120 x y^{4} + 6048 y^{5} (3 x)^{2} + 1344 y^{6} 3 x + 128 y^{7} : 2187 x^{\frac{7}{3}} + 10206 x^{2} y + 20412 y^{2} x^{\frac{5}{3}} + 22680 y^{3} x^{\frac{4}{3}} + 15120 x y^{4} + 6048 y^{5} (3 x)^{2} + 1344 y^{6} 3 x + 128 y^{7}

= 2187 x^{\frac{7}{3}} + 10206 x^{2} y + 20412 y^{2} x^{\frac{5}{3}} + 22680 y^{3} x^{\frac{4}{3}} + 15120 x y^{4} + 6048 y^{5} (3 x)^{2} + 1344 y^{6} 3 x + 128 y^{7}

Semplificare

2187 x^{\frac{7}{3}} + 10206 x^{2} y + 20412 y^{2} x^{\frac{5}{3}} + 22680 y^{3} x^{\frac{4}{3}} + 15120 x y^{4} + 6048 y^{5} (3 x)^{2} + 1344 y^{6} 3 x + 128 y^{7} : 2187 3 x^{7} + 10206 x^{2} y + 20412 y^{2} 3 x^{5} + 22680 y^{3} 3 x^{4} + 15120 x y^{4} + 6048 y^{5} (3 x)^{2} + 1344 y^{6} 3 x + 128 y^{7}

= 2187 3 x^{7} + 10206 x^{2} y + 20412 y^{2} 3 x^{5} + 22680 y^{3} 3 x^{4} + 15120 x y^{4} + 6048 y^{5} (3 x)^{2} + 1344 y^{6} 3 x + 128 y^{7}

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