Question

Find the square root of
$\sqrt{7 + 2 \sqrt{6} }$
​

Answer 1

Given:

The expression -

$\bf \dashrightarrow \sqrt{7 + 2\sqrt{6}}$

What To Find:

We have to find -

• The square root of the expression.

How To Find:

To find, we have to -

• First, we have to find it in the form of √a + √b.
• Second. we can take a = 7 and b = 2√6.
• Third, square both sides.
• Fourth, use the identities to simplify it.
• Finally, find the square root of the expression.

Solution:

Let's take the form of √a + √b, where a = 7 and b = 2√6.

Also written as,

$\sf \implies \sqrt{a} + \sqrt{b} =\sqrt{ 7 + 2\sqrt{6}}$

Squaring both sides,

$\sf \implies (\sqrt{a} + \sqrt{b})^2 = \bigg( \sqrt{ 7 + 2\sqrt{6}} \bigg) ^2$

Use the identity (√a + √b)² = a + b - 2√(ab),

$\sf \implies a + b - 2\sqrt{ab} = 7 + 2\sqrt{6}$

Where,

• a + b = 7
• 2√(ab) = 2√6

Cancel 2 in 2√(ab) and 2√6,

• ab = 6

By manual guessing we can see that,

• a + b = 6 + 1 = 7
• ab = 6 × 1 = 6

That is,

• a = 6
• b = 1

Write it in the form of √a + √b,

$\sf \implies \sqrt{6} + \sqrt{1}$

Here √1 = 1, so,

$\sf \implies \sqrt{6} + 1$

Final Answer:

∴ Thus, the square root of √[7 + 2√6] is √6 + 1.