What is expand (5+e^{3x})^5 ?

The solution to expand (5+e^{3x})^5 is 3125+3125e^{3x}+1250e^{6x}+250e^{9x}+25e^{12x}+e^{15x}

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expand (5+e^{3x})^5

expand

(5 + e^{3 x})^{5}

3125 + 3125 e^{3 x} + 1250 e^{6 x} + 250 e^{9 x} + 25 e^{12 x} + e^{15 x}

Solution steps

(5 + e^{3 x})^{5}

Apply binomial theorem:

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

a = 5, b = e^{3 x}

= i = 0 \sum 5 (i 5) \cdot 5^{(5 - i)} (e^{3 x})^{i}

Expand summation

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

\frac{5 !}{0 ! ( 5 - 0 ) !} \cdot 5^{5} (e^{3 x})^{0} = 3125

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} \cdot 5^{4} (e^{3 x})^{1} : 3125 e^{3 x}

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} \cdot 5^{3} (e^{3 x})^{2} : 1250 e^{6 x}

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} \cdot 5^{2} (e^{3 x})^{3} : 250 e^{9 x}

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} \cdot 5^{1} (e^{3 x})^{4} : 25 e^{12 x}

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} \cdot 5^{0} (e^{3 x})^{5} : e^{15 x}

= 3125 + 3125 e^{3 x} + 1250 e^{6 x} + 250 e^{9 x} + 25 e^{12 x} + e^{15 x}

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What is expand (5+e^{3x})^5 ?
The solution to expand (5+e^{3x})^5 is 3125+3125e^{3x}+1250e^{6x}+250e^{9x}+25e^{12x}+e^{15x}