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Popular Algebra >

expand (2x^3+3y^2)^5

Solution

expand

(2 x^{3} + 3 y^{2})^{5}

Solution

32 x^{15} + 240 x^{12} y^{2} + 720 x^{9} y^{4} + 1080 x^{6} y^{6} + 810 x^{3} y^{8} + 243 y^{10}

Solution steps

(2 x^{3} + 3 y^{2})^{5}

Apply binomial theorem:

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

a = 2 x^{3}, b = 3 y^{2}

= i = 0 \sum 5 (i 5) (2 x^{3})^{(5 - i)} (3 y^{2})^{i}

Expand summation

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

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} (2 x^{3})^{5} (3 y^{2})^{0} : 32 x^{15}

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} (2 x^{3})^{4} (3 y^{2})^{1} : 240 x^{12} y^{2}

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} (2 x^{3})^{3} (3 y^{2})^{2} : 720 x^{9} y^{4}

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} (2 x^{3})^{2} (3 y^{2})^{3} : 1080 x^{6} y^{6}

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} (2 x^{3})^{1} (3 y^{2})^{4} : 810 x^{3} y^{8}

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} (2 x^{3})^{0} (3 y^{2})^{5} : 243 y^{10}

= 32 x^{15} + 240 x^{12} y^{2} + 720 x^{9} y^{4} + 1080 x^{6} y^{6} + 810 x^{3} y^{8} + 243 y^{10}

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