Consider the algebraic expression E=(2x+y) (5a-b)^3

To expand and simplify the given algebraic expression E=(2x+y)(5a-b)^3, follow these steps:

Step 1: Apply the exponent to the binomial (5a-b)^3 using the binomial theorem. Recall that the binomial theorem states that (a+b)^n = C(n,0)a^n*b^0 + C(n,1)a^(n-1)b^1 + ... + C(n,n-1)a^1b^(n-1) + C(n,n)a^0b^n, where C(n,k) represents the binomial coefficient or "n choose k".

In this case, n = 3 (the exponent), a = 5a, and b = -b. Applying the binomial theorem, we have:

(5a-b)^3 = C(3,0)(5a)^3*(-b)^0 + C(3,1)(5a)^2*(-b)^1 + C(3,2)(5a)^1*(-b)^2 + C(3,3)(5a)^0*(-b)^3
= 1*(5a)^3*(-b)^0 + 3*(5a)^2*(-b)^1 + 3*(5a)^1*(-b)^2 + 1*(5a)^0*(-b)^3
= (5a)^3 + 3*(5a)^2*(-b) + 3*(5a)*(-b)^2 + (-b)^3
= 125a^3 - 75a^2b + 15ab^2 - b^3

Step 2: Substitute the simplified expression obtained in Step 1 back into the original expression E=(2x+y)(5a-b)^3.

E = (2x + y)(125a^3 - 75a^2b + 15ab^2 - b^3)

So, the expanded and simplified expression is E = (2x + y)(125a^3 - 75a^2b + 15ab^2 - b^3).