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Popular Algebra >

expand (2x+3y)^7

Solution

expand

(2 x + 3 y)^{7}

Solution

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

Solution steps

(2 x + 3 y)^{7}

Apply binomial theorem:

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

a = 2 x, b = 3 y

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

Expand summation

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

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

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