Question

Show that (2 + √3) (2-√3) (5+√2) (5-√2)

Answer 1

Step-by-step explanation:

To prove: 3 + 2√5 is an irrational number.

Proof:

Let us assume that 3 + 2√5 is a rational number.

So, it can be written in the form a/b

3 + 2√5 = a/b

Here a and b are coprime numbers and b ≠ 0

Solving 3 + 2√5 = a/b we get,

=>2√5 = a/b – 3

=>2√5 = (a-3b)/b

=>√5 = (a-3b)/2b

This shows (a-3b)/2b is a rational number. But we know that √5 is an irrational number.

So, it contradicts our assumption. Our assumption of 3 + 2√5 is a rational number is incorrect.

3 + 2√5 is an irrational number

Hence proved

Answer 2

Step-by-step explanation:

${\huge{\bf{\fbox{\orange{Solution :→}}}}}$

(2 + √3) (2 - √3) (5 + √2) (5 - √2)

${\bold{\sf{\fbox{\pink{✥ \: Let's apply (a-b)(a+b) = a²-b²}}}}}$

${\large{\sf{:⇒ \: (2 + √3) \: (2 - √3) \: (5 + √2) \: (5 - √2)}}}$

${\large{\sf{:⇒ \: (4 - 3) \: (25 - 2)}}}$

${\large{\sf{:⇒ \: 1 \: × \: 23}}}$

${\large{\sf{\fbox{\green{⇢ \: 23}}}}}$

Hence, 23 is a rational number