What is expand (1/2*sqrt(x)-1/2)^{10} ?

The solution to expand (1/2*sqrt(x)-1/2)^{10} is

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expand (1/2*sqrt(x)-1/2)^{10}

expand

(\frac{1}{2} \cdot x - \frac{1}{2})^{10}

\frac{x ^{5}}{1024} + \frac{- 5 x ^{4} x + 105 x ^{3} + 105 x ^{2} - 5 x}{512} + \frac{45 x ^{4}}{1024} + \frac{- 15 x ^{3} x - 15 x x}{128} - \frac{63 x ^{2} x}{256} + \frac{45 x}{1024} + \frac{1}{1024}

Solution steps

(\frac{1}{2} x - \frac{1}{2})^{10}

Apply binomial theorem:

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

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

= i = 0 \sum 10 (i 10) (\frac{1}{2} x)^{(10 - i)} (- \frac{1}{2})^{i}

Expand summation

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

Simplify

\frac{10 !}{0 ! ( 10 - 0 ) !} (\frac{1}{2} x)^{10} (- \frac{1}{2})^{0} : \frac{x ^{5}}{1024}

Simplify

\frac{10 !}{1 ! ( 10 - 1 ) !} (\frac{1}{2} x)^{9} (- \frac{1}{2})^{1} : - \frac{5 ( x ) ^{9}}{512}

Simplify

\frac{10 !}{2 ! ( 10 - 2 ) !} (\frac{1}{2} x)^{8} (- \frac{1}{2})^{2} : \frac{45 x ^{4}}{1024}

Simplify

\frac{10 !}{3 ! ( 10 - 3 ) !} (\frac{1}{2} x)^{7} (- \frac{1}{2})^{3} : - \frac{15 ( x ) ^{7}}{128}

Simplify

\frac{10 !}{4 ! ( 10 - 4 ) !} (\frac{1}{2} x)^{6} (- \frac{1}{2})^{4} : \frac{105 x ^{3}}{512}

Simplify

\frac{10 !}{5 ! ( 10 - 5 ) !} (\frac{1}{2} x)^{5} (- \frac{1}{2})^{5} : - \frac{63 ( x ) ^{5}}{256}

Simplify

\frac{10 !}{6 ! ( 10 - 6 ) !} (\frac{1}{2} x)^{4} (- \frac{1}{2})^{6} : \frac{105 x ^{2}}{512}

Simplify

\frac{10 !}{7 ! ( 10 - 7 ) !} (\frac{1}{2} x)^{3} (- \frac{1}{2})^{7} : - \frac{15 ( x ) ^{3}}{128}

Simplify

\frac{10 !}{8 ! ( 10 - 8 ) !} (\frac{1}{2} x)^{2} (- \frac{1}{2})^{8} : \frac{45 x}{1024}

Simplify

\frac{10 !}{9 ! ( 10 - 9 ) !} (\frac{1}{2} x)^{1} (- \frac{1}{2})^{9} : - \frac{5 x}{512}

Simplify

\frac{10 !}{10 ! ( 10 - 10 ) !} (\frac{1}{2} x)^{0} (- \frac{1}{2})^{10} : \frac{1}{1024}

= \frac{x ^{5}}{1024} - \frac{5 ( x ) ^{9}}{512} + \frac{45 x ^{4}}{1024} - \frac{15 ( x ) ^{7}}{128} + \frac{105 x ^{3}}{512} - \frac{63 ( x ) ^{5}}{256} + \frac{105 x ^{2}}{512} - \frac{15 ( x ) ^{3}}{128} + \frac{45 x}{1024} - \frac{5 x}{512} + \frac{1}{1024}

Simplify

\frac{x ^{5}}{1024} - \frac{5 ( x ) ^{9}}{512} + \frac{45 x ^{4}}{1024} - \frac{15 ( x ) ^{7}}{128} + \frac{105 x ^{3}}{512} - \frac{63 ( x ) ^{5}}{256} + \frac{105 x ^{2}}{512} - \frac{15 ( x ) ^{3}}{128} + \frac{45 x}{1024} - \frac{5 x}{512} + \frac{1}{1024} : \frac{x ^{5}}{1024} + \frac{- 5 x ^{4} x + 105 x ^{3} + 105 x ^{2} - 5 x}{512} + \frac{45 x ^{4}}{1024} + \frac{- 15 x ^{3} x - 15 x x}{128} - \frac{63 x ^{2} x}{256} + \frac{45 x}{1024} + \frac{1}{1024}

= \frac{x ^{5}}{1024} + \frac{- 5 x ^{4} x + 105 x ^{3} + 105 x ^{2} - 5 x}{512} + \frac{45 x ^{4}}{1024} + \frac{- 15 x ^{3} x - 15 x x}{128} - \frac{63 x ^{2} x}{256} + \frac{45 x}{1024} + \frac{1}{1024}

x^{\frac{3}{5}} (x - 8)

\frac{x}{( 0.1111 - 2 x ) ^{2}}

lo g_{8} (64)

\frac{8 x}{( x + 4 ) ( x - 5 )}

(1 + tan (\frac{x}{2}))^{2}

What is expand (1/2*sqrt(x)-1/2)^{10} ?
The solution to expand (1/2*sqrt(x)-1/2)^{10} is