What is expand (x+sqrt(x))^6 ?

The solution to expand (x+sqrt(x))^6 is x^6+6sqrt(x^{11)}+15x^5+20sqrt(x^9)+15x^4+6sqrt(x^7)+x^3

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expand (x+sqrt(x))^6

expand

(x + x)^{6}

x^{6} + 6 x^{11} + 15 x^{5} + 20 x^{9} + 15 x^{4} + 6 x^{7} + x^{3}

Solution steps

(x + x)^{6}

Apply binomial theorem:

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

a = x, b = x

= i = 0 \sum 6 (i 6) x^{(6 - i)} (x)^{i}

Expand summation

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

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} x^{6} (x)^{0} : x^{6}

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} x^{5} (x)^{1} : 6 x^{5} x

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} x^{4} (x)^{2} : 15 x^{5}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} x^{3} (x)^{3} : 20 x^{3} (x)^{3}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} x^{2} (x)^{4} : 15 x^{4}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} x^{1} (x)^{5} : 6 x (x)^{5}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} x^{0} (x)^{6} : x^{3}

= x^{6} + 6 x^{5} x + 15 x^{5} + 20 x^{3} (x)^{3} + 15 x^{4} + 6 x (x)^{5} + x^{3}

Simplify

x^{6} + 6 x^{5} x + 15 x^{5} + 20 x^{3} (x)^{3} + 15 x^{4} + 6 x (x)^{5} + x^{3} : x^{6} + 6 x^{\frac{11}{2}} + 15 x^{5} + 20 x^{\frac{9}{2}} + 15 x^{4} + 6 x^{\frac{7}{2}} + x^{3}

= x^{6} + 6 x^{\frac{11}{2}} + 15 x^{5} + 20 x^{\frac{9}{2}} + 15 x^{4} + 6 x^{\frac{7}{2}} + x^{3}

Simplify

x^{6} + 6 x^{\frac{11}{2}} + 15 x^{5} + 20 x^{\frac{9}{2}} + 15 x^{4} + 6 x^{\frac{7}{2}} + x^{3} : x^{6} + 6 x^{11} + 15 x^{5} + 20 x^{9} + 15 x^{4} + 6 x^{7} + x^{3}

= x^{6} + 6 x^{11} + 15 x^{5} + 20 x^{9} + 15 x^{4} + 6 x^{7} + x^{3}

(1 + e^{- j x}) (1 + e^{+ j x})

x ln (85) + (1 - x) ln (10)

(x - \frac{1}{x}) sin (θ)

(x - h)^{2} + (y - k)^{2} + (z - l)^{2^{^{'}}}

(\frac{1}{2} - z^{2}) (\frac{1}{2} + z)

What is expand (x+sqrt(x))^6 ?
The solution to expand (x+sqrt(x))^6 is x^6+6sqrt(x^{11)}+15x^5+20sqrt(x^9)+15x^4+6sqrt(x^7)+x^3