Solve (2a^x+1/2b^x) ^2

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