Expand Using the Binomial Theorem (x^2+y^3)^4

Problem

(x2+y3)4

Solution

1. Identify the components of the binomial (a+b)n where a=x2 b=y3 and n=4

2. Apply the Binomial Theorem formula, which states (a+b)n=(∑_k=0^n)((n/k))*a(n−k)*bk)

3. Write the expansion by substituting the values into the summation for k=0,1,2,3,4

(4/0)(x2)4*(y3)0+(4/1)(x2)3*(y3)1+(4/2)(x2)2*(y3)2+(4/3)(x2)1*(y3)3+(4/4)(x2)0*(y3)4)))))

4. Calculate the binomial coefficients using (n/k)=(n!)/(k!(n−k)!)) which results in 1, 4, 6, 4, 1$.

5. Simplify the powers using the rule (xa)b=x(a*b)

1*(x8)*(1)+4*(x6)*(y3)+6*(x4)*(y6)+4*(x2)*(y9)+1*(1)*(y12)

6. Combine the terms to reach the final expanded form.

x8+4*x6*y3+6*x4*y6+4*x2*y9+y12

Final Answer

(x2+y3)4=x8+4*x6*y3+6*x4*y6+4*x2*y9+y12

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