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Popular Algebra >

expand (1+2x)^{11}

Solution

expand

(1 + 2 x)^{11}

Solution

1 + 22 x + 220 x^{2} + 1320 x^{3} + 5280 x^{4} + 14784 x^{5} + 29568 x^{6} + 42240 x^{7} + 42240 x^{8} + 28160 x^{9} + 11264 x^{10} + 2048 x^{11}

Solution steps

(1 + 2 x)^{11}

Apply binomial theorem:

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

a = 1, b = 2 x

= i = 0 \sum 11 (i 11) \cdot 1^{(11 - i)} (2 x)^{i}

Expand summation

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

Simplify

\frac{11 !}{0 ! ( 11 - 0 ) !} \cdot 1^{11} (2 x)^{0} : 1

Simplify

\frac{11 !}{1 ! ( 11 - 1 ) !} \cdot 1^{10} (2 x)^{1} : 22 x

Simplify

\frac{11 !}{2 ! ( 11 - 2 ) !} \cdot 1^{9} (2 x)^{2} : 220 x^{2}

Simplify

\frac{11 !}{3 ! ( 11 - 3 ) !} \cdot 1^{8} (2 x)^{3} : 1320 x^{3}

Simplify

\frac{11 !}{4 ! ( 11 - 4 ) !} \cdot 1^{7} (2 x)^{4} : 5280 x^{4}

Simplify

\frac{11 !}{5 ! ( 11 - 5 ) !} \cdot 1^{6} (2 x)^{5} : 14784 x^{5}

Simplify

\frac{11 !}{6 ! ( 11 - 6 ) !} \cdot 1^{5} (2 x)^{6} : 29568 x^{6}

Simplify

\frac{11 !}{7 ! ( 11 - 7 ) !} \cdot 1^{4} (2 x)^{7} : 42240 x^{7}

Simplify

\frac{11 !}{8 ! ( 11 - 8 ) !} \cdot 1^{3} (2 x)^{8} : 42240 x^{8}

Simplify

\frac{11 !}{9 ! ( 11 - 9 ) !} \cdot 1^{2} (2 x)^{9} : 28160 x^{9}

Simplify

\frac{11 !}{10 ! ( 11 - 10 ) !} \cdot 1^{1} (2 x)^{10} : 11264 x^{10}

Simplify

\frac{11 !}{11 ! ( 11 - 11 ) !} \cdot 1^{0} (2 x)^{11} : 2048 x^{11}

= 1 + 22 x + 220 x^{2} + 1320 x^{3} + 5280 x^{4} + 14784 x^{5} + 29568 x^{6} + 42240 x^{7} + 42240 x^{8} + 28160 x^{9} + 11264 x^{10} + 2048 x^{11}

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