If (2⁵ + 0.125)² - (2⁵ - 0.125)² = 2^x, then the value of x is :

1. 2

2. 3

3. 4

4. 5

Given,

(2⁵ + 0.125)² - (2⁵ - 0.125)² = 2^x

By using the identity:

(a + b)² - (a - b)² = 4ab

Let a = 2⁵, b = 0.125

Using identity in L.H.S. of the given equation,

$\Rightarrow (2^5 + 0.125)^2 - (2^5 - 0.125)^2 = 4 \times 2^5 \times 0.125 \\[1em] = 4 \times 32 \times \dfrac{125}{1000} \\[1em] = 4 \times 32 \times \dfrac{1}{8} \\[1em] = 16 \\[1em] = 2^4.$

Equation L.H.S. and R.H.S.,

⇒ 2^x = 2⁴

⇒ x = 4.

Hence, option 3 is the correct option.