Simplify 1/2b^x to b^x/2
(2a^x + b^x/2)^2
Use the Square of Sum: (a + b)^2 = a^2 + 2ab + b^2
(2a^x)^2 + 2 × 2a^x × b^x/2 + (b^x/2)^2
Use the Multiplication Distributive Property: (xy)^a = x^a y^a
2^2(a^x)^2 + 2 ×2a^x × b^x/2 + (b^x/2)^2
Simplify 2^2 to 4
4(a^x) + 2 × 2a^x × b^x/2 + (b^x/2)^2
Use the Power Rule: (x^a)^b = x^ab
4a^x × 2 + 2 × 2a^x × b^x/2 + (b^x/2)^2
Regroup terms
4a^2x + 2 × 2a^x × b^x/2 + (b^x/2)^2
Use Division Distributive Property: (x/y)^a = x^a/y^a
4a^2x + 2 × 2a^x × b^x/2 + (b^x)^2/2^2
Use the Power Rule: (x^a)^b = x^ab
4a^2x + 2 × 2a^x × b^x/2 + b^x × 2/2^2
Regroup terms
4a^2x + 2 ×2a^x × b^x/2 + b^2x/2^2
Simplify 2^2 to 4
4a^2x + 2 × 2a^x × b^x/2 + b^2x/4
Simplify 2 × 2a^x × b^x/2 to 4a^xb^x/2
4a^2x + 4a^xb^x/2 + b^2x/4
Simplify 4a^xb^x/2 to 2a^xb^x
= 4a^2x + 2a^xb^x + b^2x/4