**Solution:**
Given, x = 2ab/b^2

We need to find the value of { (a x) (a-×) ÷ (a ×) - (a-×) } ^1/2

Let's simplify the given expression step by step.

**Step 1: Find the value of a-×**

a-× = 1/a

**Step 2: Find the value of (a x)**

(a x) = a * 2ab/b^2 = 2a^2/b

**Step 3: Find the value of (a x) - (a-×)**

(a x) - (a-×) = 2a^2/b - 1/a

Taking LCM, we get

((a x) - (a-×)) = (2a^3 - b)/ab

**Step 4: Find the value of (a x) (a-×)**

(a x) (a-×) = (2ab/b^2) * (1/a) = 2a/b

**Step 5: Substituting the values in the given expression**

{ (a x) (a-×) ÷ (a x) - (a-×) } ^1/2

= { (2a/b) ÷ ((2a^3 - b)/ab) } ^1/2

= { 2ab ÷ (2a^3 - b) } ^1/2

= { 2ab ÷ [(a√2)^(3) - b] } ^1/2

= { 2ab ÷ [(a√2 - b)(a^2 + ab√2 + b^2)] } ^1/2

= { 2ab ÷ [(a√2 - b)(a√2 - b) (a + b√2)] } ^1/2

= { 2ab ÷ [(a√2 - b)^2 (a + b√2)] } ^1/2

= { 2ab ÷ [a^3 + 2a^2b√2 - ab^2√2 - b^3] } ^1/2

Hence, the value of the given expression is { 2ab ÷ [a^3 + 2a^2b√2 - ab^2√2 - b^3] } ^1/2.

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