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Popular Algebra >

expand (1+dx)^{10}

Solution

expand

(1 + d x)^{10}

Solution

1 + 10 d x + 45 d x^{2} + 120 d x^{3} + 210 d x^{4} + 252 d x^{5} + 210 d x^{6} + 120 d x^{7} + 45 d x^{8} + 10 d x^{9} + d x^{10}

Solution steps

(1 + d x)^{10}

Apply binomial theorem:

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

a = 1, b = d x

= i = 0 \sum 10 (i 10) \cdot 1^{(10 - i)} d x^{i}

Expand summation

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

\frac{10 !}{0 ! ( 10 - 0 ) !} \cdot 1^{10} d x^{0} = 1

Simplify

\frac{10 !}{1 ! ( 10 - 1 ) !} \cdot 1^{9} d x^{1} : 10 d x

Simplify

\frac{10 !}{2 ! ( 10 - 2 ) !} \cdot 1^{8} d x^{2} : 45 d x^{2}

Simplify

\frac{10 !}{3 ! ( 10 - 3 ) !} \cdot 1^{7} d x^{3} : 120 d x^{3}

Simplify

\frac{10 !}{4 ! ( 10 - 4 ) !} \cdot 1^{6} d x^{4} : 210 d x^{4}

Simplify

\frac{10 !}{5 ! ( 10 - 5 ) !} \cdot 1^{5} d x^{5} : 252 d x^{5}

Simplify

\frac{10 !}{6 ! ( 10 - 6 ) !} \cdot 1^{4} d x^{6} : 210 d x^{6}

Simplify

\frac{10 !}{7 ! ( 10 - 7 ) !} \cdot 1^{3} d x^{7} : 120 d x^{7}

Simplify

\frac{10 !}{8 ! ( 10 - 8 ) !} \cdot 1^{2} d x^{8} : 45 d x^{8}

Simplify

\frac{10 !}{9 ! ( 10 - 9 ) !} \cdot 1^{1} d x^{9} : 10 d x^{9}

Simplify

\frac{10 !}{10 ! ( 10 - 10 ) !} \cdot 1^{0} d x^{10} : d x^{10}

= 1 + 10 d x + 45 d x^{2} + 120 d x^{3} + 210 d x^{4} + 252 d x^{5} + 210 d x^{6} + 120 d x^{7} + 45 d x^{8} + 10 d x^{9} + d x^{10}

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