What is expand (x+y)^8 ?

The solution to expand (x+y)^8 is x^8+8x^7y+28x^6y^2+56x^5y^3+70x^4y^4+56x^3y^5+28x^2y^6+8xy^7+y^8

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expand (x+y)^8

expand

(x + y)^{8}

x^{8} + 8 x^{7} y + 28 x^{6} y^{2} + 56 x^{5} y^{3} + 70 x^{4} y^{4} + 56 x^{3} y^{5} + 28 x^{2} y^{6} + 8 x y^{7} + y^{8}

Solution steps

(x + y)^{8}

Apply binomial theorem:

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

a = x, b = y

= i = 0 \sum 8 (i 8) x^{(8 - i)} y^{i}

Expand summation

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

Simplify

\frac{8 !}{0 ! ( 8 - 0 ) !} x^{8} y^{0} : x^{8}

Simplify

\frac{8 !}{1 ! ( 8 - 1 ) !} x^{7} y^{1} : 8 x^{7} y

Simplify

\frac{8 !}{2 ! ( 8 - 2 ) !} x^{6} y^{2} : 28 x^{6} y^{2}

Simplify

\frac{8 !}{3 ! ( 8 - 3 ) !} x^{5} y^{3} : 56 x^{5} y^{3}

Simplify

\frac{8 !}{4 ! ( 8 - 4 ) !} x^{4} y^{4} : 70 x^{4} y^{4}

Simplify

\frac{8 !}{5 ! ( 8 - 5 ) !} x^{3} y^{5} : 56 x^{3} y^{5}

Simplify

\frac{8 !}{6 ! ( 8 - 6 ) !} x^{2} y^{6} : 28 x^{2} y^{6}

Simplify

\frac{8 !}{7 ! ( 8 - 7 ) !} x^{1} y^{7} : 8 x y^{7}

Simplify

\frac{8 !}{8 ! ( 8 - 8 ) !} x^{0} y^{8} : y^{8}

= x^{8} + 8 x^{7} y + 28 x^{6} y^{2} + 56 x^{5} y^{3} + 70 x^{4} y^{4} + 56 x^{3} y^{5} + 28 x^{2} y^{6} + 8 x y^{7} + y^{8}

\frac{sin ( x )}{1 + cos ( x )} + \frac{1 + cos ( x )}{sin ( x )}

(x + 4) (x + 3)

x \cdot 2 x

\frac{3 x}{x}

\frac{6}{2}

What is expand (x+y)^8 ?
The solution to expand (x+y)^8 is x^8+8x^7y+28x^6y^2+56x^5y^3+70x^4y^4+56x^3y^5+28x^2y^6+8xy^7+y^8