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Popular >

expand (1/2+xy)^7

Solution

expand

(\frac{1}{2} + x y)^{7}

Solution

\frac{1}{128} + \frac{7 x y}{64} + \frac{21 x ^{2} y ^{2}}{32} + \frac{35 x ^{3} y ^{3}}{16} + \frac{35 x ^{4} y ^{4}}{8} + \frac{21 x ^{5} y ^{5}}{4} + \frac{7 x ^{6} y ^{6}}{2} + x^{7} y^{7}

Solution steps

(\frac{1}{2} + x y)^{7}

Apply binomial theorem:

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

a = \frac{1}{2}, b = x y

= i = 0 \sum 7 (i 7) (\frac{1}{2})^{(7 - i)} (x y)^{i}

Expand summation

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

Simplify

\frac{7 !}{0 ! ( 7 - 0 ) !} (\frac{1}{2})^{7} (x y)^{0} : \frac{1}{128}

Simplify

\frac{7 !}{1 ! ( 7 - 1 ) !} (\frac{1}{2})^{6} (x y)^{1} : \frac{7 x y}{64}

Simplify

\frac{7 !}{2 ! ( 7 - 2 ) !} (\frac{1}{2})^{5} (x y)^{2} : \frac{21 x ^{2} y ^{2}}{32}

Simplify

\frac{7 !}{3 ! ( 7 - 3 ) !} (\frac{1}{2})^{4} (x y)^{3} : \frac{35 x ^{3} y ^{3}}{16}

Simplify

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

Simplify

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

Simplify

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

Simplify

\frac{7 !}{7 ! ( 7 - 7 ) !} (\frac{1}{2})^{0} (x y)^{7} : x^{7} y^{7}

= \frac{1}{128} + \frac{7 x y}{64} + \frac{21 x ^{2} y ^{2}}{32} + \frac{35 x ^{3} y ^{3}}{16} + \frac{35 x ^{4} y ^{4}}{8} + \frac{21 x ^{5} y ^{5}}{4} + \frac{7 x ^{6} y ^{6}}{2} + x^{7} y^{7}

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Frequently Asked Questions (FAQ)

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