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

Problem

(x^2+y^3)^4

Solution

1. Identify the components of the binomial (a+b)^n where a=x^2 b=y^3 and n=4

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

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

(4/0)(x^2)^4*(y^3)^0+(4/1)(x^2)^3*(y^3)^1+(4/2)(x^2)^2*(y^3)^2+(4/3)(x^2)^1*(y^3)^3+(4/4)(x^2)^0*(y^3)^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 (x^a)^b=x^(a*b)

1*(x^8)*(1)+4*(x^6)*(y^3)+6*(x^4)*(y^6)+4*(x^2)*(y^9)+1*(1)*(y^12)

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

x^8+4*x^6*y^3+6*x^4*y^6+4*x^2*y^9+y^12

Final Answer

(x^2+y^3)^4=x^8+4*x^6*y^3+6*x^4*y^6+4*x^2*y^9+y^12

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