Discussion

Explanation:

√128 - √56 = √2 × (√64 - √28) = √2 × (8 - 2√7)

Now, square root of √2 is $\sqrt[4]{2}$

Let us calculate square root of (8 - 2√7)

Square root of 8 - 2√7 will be of the form √x - √y.

∴ (√x - √y)2 = 8 - 2√7

⇒ x + y - 2√xy = 8 - 2√7

Now, rational part of RHS = rational part on LHS
⇒ x + y = 8   …(1)

Also, irrational part of RHS = irrational part on LHS
⇒ - 2√xy = - 2√7
⇒ xy = 7   …(2)

Solving (1) and (2), we get
x = 1 and y = 7 or x = 7 and y = 1.

The square root of 8 - 2√7 will be either √7 - 1 or 1 - √7

We will reject 1 - √7 as it is negative

Square root of √128 - √56 = $\sqrt[4]{2}\times \left(\sqrt{7}-1\right)$

Hence, option (a).

» Your doubt will be displayed only after approval.