Question

prove that : √23+√528=2√3+√11

Answer 1

Question:

$\text {Prove that }\sqrt{23 + \sqrt{528} } = 2 \sqrt{3 } + \sqrt{11}$

Proof:

$\text{RHS = }2 \sqrt{3 } + \sqrt{11}$

Square the RHS:

$\text{RHS}^2 = (2 \sqrt{3 } + \sqrt{11})^2$

Apply (a + b)² = a² + 2ab + b²:

$\text{RHS}^2 = (2 \sqrt{3 })^2 + 2(2 \sqrt{3 }) (\sqrt{11}) + (\sqrt{11})^2$

$\text{RHS}^2 =4(3) + 4 (\sqrt{3}) (\sqrt{11}) + 11$

$\text{RHS}^2 = 23 + 4(\sqrt{33})$

Write 4 as √16:

$\text{RHS}^2 = 23 + \sqrt{16} (\sqrt{33})$

$\text{RHS}^2 = 23 + \sqrt{528}$

Square root it:

$\text{RHS} = \sqrt{23 + \sqrt{528}}$

$\text{RHS} = \text{LHS}$

Answer 2

Given:

√23 + √528

To prove:

√23 + √528 = 2√3 + √11

Solution:

Consider,

2√3 + √11

Squaring,

( 2√3 + √11 )^2

This is of the algebraic expression ( a + b )^2

( a + b )^2 = a^2 + 2 ( a ) ( b ) + b^2

Substituing,

We get,

( 2√3 )^2 + 2 ( 2√3 ) ( √11 ) + ( √11 )^2

4 ( 3 ) + 4 ( √3 ) ( √11 ) + 11

23 + 4√33

Here,

4 = √16

Substituting,

We get,

23 + √16 * √33

23 + √528

Taking square root on both sides,

√23+√528

Hence, √23 + √528 = 2√3 + √11