What is expand (1-w^2)^7 ?

The solution to expand (1-w^2)^7 is 1-7w^2+21w^4-35w^6+35w^8-21w^{10}+7w^{12}-w^{14}

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expand (1-w^2)^7

expand

(1 - w^{2})^{7}

1 - 7 w^{2} + 21 w^{4} - 35 w^{6} + 35 w^{8} - 21 w^{10} + 7 w^{12} - w^{14}

Solution steps

(1 - w^{2})^{7}

Apply binomial theorem:

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

a = 1, b = - w^{2}

= i = 0 \sum 7 (i 7) \cdot 1^{(7 - i)} (- w^{2})^{i}

Expand summation

= \frac{7 !}{0 ! ( 7 - 0 ) !} \cdot 1^{7} (- w^{2})^{0} + \frac{7 !}{1 ! ( 7 - 1 ) !} \cdot 1^{6} (- w^{2})^{1} + \frac{7 !}{2 ! ( 7 - 2 ) !} \cdot 1^{5} (- w^{2})^{2} + \frac{7 !}{3 ! ( 7 - 3 ) !} \cdot 1^{4} (- w^{2})^{3} + \frac{7 !}{4 ! ( 7 - 4 ) !} \cdot 1^{3} (- w^{2})^{4} + \frac{7 !}{5 ! ( 7 - 5 ) !} \cdot 1^{2} (- w^{2})^{5} + \frac{7 !}{6 ! ( 7 - 6 ) !} \cdot 1^{1} (- w^{2})^{6} + \frac{7 !}{7 ! ( 7 - 7 ) !} \cdot 1^{0} (- w^{2})^{7}

Simplify

\frac{7 !}{0 ! ( 7 - 0 ) !} \cdot 1^{7} (- w^{2})^{0} : 1

Simplify

\frac{7 !}{1 ! ( 7 - 1 ) !} \cdot 1^{6} (- w^{2})^{1} : - 7 w^{2}

Simplify

\frac{7 !}{2 ! ( 7 - 2 ) !} \cdot 1^{5} (- w^{2})^{2} : 21 w^{4}

Simplify

\frac{7 !}{3 ! ( 7 - 3 ) !} \cdot 1^{4} (- w^{2})^{3} : - 35 w^{6}

Simplify

\frac{7 !}{4 ! ( 7 - 4 ) !} \cdot 1^{3} (- w^{2})^{4} : 35 w^{8}

Simplify

\frac{7 !}{5 ! ( 7 - 5 ) !} \cdot 1^{2} (- w^{2})^{5} : - 21 w^{10}

Simplify

\frac{7 !}{6 ! ( 7 - 6 ) !} \cdot 1^{1} (- w^{2})^{6} : 7 w^{12}

Simplify

\frac{7 !}{7 ! ( 7 - 7 ) !} \cdot 1^{0} (- w^{2})^{7} : - w^{14}

= 1 - 7 w^{2} + 21 w^{4} - 35 w^{6} + 35 w^{8} - 21 w^{10} + 7 w^{12} - w^{14}

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What is expand (1-w^2)^7 ?
The solution to expand (1-w^2)^7 is 1-7w^2+21w^4-35w^6+35w^8-21w^{10}+7w^{12}-w^{14}