What is expand (x+3y)^4 ?

The solution to expand (x+3y)^4 is x^4+12x^3y+54x^2y^2+108xy^3+81y^4

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expand (x+3y)^4

expand

(x + 3 y)^{4}

x^{4} + 12 x^{3} y + 54 x^{2} y^{2} + 108 x y^{3} + 81 y^{4}

Solution steps

(x + 3 y)^{4}

Apply binomial theorem:

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

a = x, b = 3 y

= i = 0 \sum 4 (i 4) x^{(4 - i)} (3 y)^{i}

Expand summation

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

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} x^{4} (3 y)^{0} : x^{4}

Simplify

\frac{4 !}{1 ! ( 4 - 1 ) !} x^{3} (3 y)^{1} : 12 x^{3} y

Simplify

\frac{4 !}{2 ! ( 4 - 2 ) !} x^{2} (3 y)^{2} : 54 x^{2} y^{2}

Simplify

\frac{4 !}{3 ! ( 4 - 3 ) !} x^{1} (3 y)^{3} : 108 x y^{3}

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} x^{0} (3 y)^{4} : 81 y^{4}

= x^{4} + 12 x^{3} y + 54 x^{2} y^{2} + 108 x y^{3} + 81 y^{4}

- \frac{1}{6} + \frac{1}{2}

(3 - 4 i) + (- 1.3 + 5 i)

\frac{8 y ^{2} + 10 y}{14 y ^{2}} - \frac{y ^{2} - 4 y}{14 y ^{2}}

2 π \cdot 5

(10 x)^{2}

What is expand (x+3y)^4 ?
The solution to expand (x+3y)^4 is x^4+12x^3y+54x^2y^2+108xy^3+81y^4