What is expand (2+xsin(3x))^{10} ?

The solution to expand (2+xsin(3x))^{10} is

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expand (2+xsin(3x))^{10}

expand

(2 + x sin (3 x))^{10}

1024 + 5120 x sin (3 x) + 11520 x^{2} sin^{2} (3 x) + 15360 x^{3} sin^{3} (3 x) + 13440 x^{4} sin^{4} (3 x) + 8064 x^{5} sin^{5} (3 x) + 3360 x^{6} sin^{6} (3 x) + 960 x^{7} sin^{7} (3 x) + 180 x^{8} sin^{8} (3 x) + 20 x^{9} sin^{9} (3 x) + x^{10} sin^{10} (3 x)

Solution steps

(2 + x sin (3 x))^{10}

Apply binomial theorem:

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

a = 2, b = x sin (3 x)

= i = 0 \sum 10 (i 10) \cdot 2^{(10 - i)} (x sin (3 x))^{i}

Expand summation

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

Simplify

\frac{10 !}{0 ! ( 10 - 0 ) !} \cdot 2^{10} (x sin (3 x))^{0} : 1024

Simplify

\frac{10 !}{1 ! ( 10 - 1 ) !} \cdot 2^{9} (x sin (3 x))^{1} : 5120 x sin (3 x)

Simplify

\frac{10 !}{2 ! ( 10 - 2 ) !} \cdot 2^{8} (x sin (3 x))^{2} : 11520 x^{2} sin^{2} (3 x)

Simplify

\frac{10 !}{3 ! ( 10 - 3 ) !} \cdot 2^{7} (x sin (3 x))^{3} : 15360 x^{3} sin^{3} (3 x)

Simplify

\frac{10 !}{4 ! ( 10 - 4 ) !} \cdot 2^{6} (x sin (3 x))^{4} : 13440 x^{4} sin^{4} (3 x)

Simplify

\frac{10 !}{5 ! ( 10 - 5 ) !} \cdot 2^{5} (x sin (3 x))^{5} : 8064 x^{5} sin^{5} (3 x)

Simplify

\frac{10 !}{6 ! ( 10 - 6 ) !} \cdot 2^{4} (x sin (3 x))^{6} : 3360 x^{6} sin^{6} (3 x)

Simplify

\frac{10 !}{7 ! ( 10 - 7 ) !} \cdot 2^{3} (x sin (3 x))^{7} : 960 x^{7} sin^{7} (3 x)

Simplify

\frac{10 !}{8 ! ( 10 - 8 ) !} \cdot 2^{2} (x sin (3 x))^{8} : 180 x^{8} sin^{8} (3 x)

Simplify

\frac{10 !}{9 ! ( 10 - 9 ) !} \cdot 2^{1} (x sin (3 x))^{9} : 20 x^{9} sin^{9} (3 x)

Simplify

\frac{10 !}{10 ! ( 10 - 10 ) !} \cdot 2^{0} (x sin (3 x))^{10} : x^{10} sin^{10} (3 x)

= 1024 + 5120 x sin (3 x) + 11520 x^{2} sin^{2} (3 x) + 15360 x^{3} sin^{3} (3 x) + 13440 x^{4} sin^{4} (3 x) + 8064 x^{5} sin^{5} (3 x) + 3360 x^{6} sin^{6} (3 x) + 960 x^{7} sin^{7} (3 x) + 180 x^{8} sin^{8} (3 x) + 20 x^{9} sin^{9} (3 x) + x^{10} sin^{10} (3 x)

(\frac{5 x}{4} - \frac{y}{2}) (\frac{5 x}{4} + \frac{y}{2})

(x + \frac{1}{2})^{3}

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

(x + 3 + 5) (x + 3 - 5)

- 25 (0.08 x + 0.04 y = 330)