Question

$\rm \bold{ѕhow \: тhat}$
$\rm \bold{ \frac{1}{3 - \sqrt{8} } - \frac{1}{ \sqrt{8} - \sqrt{7} } + \frac{1}{ \sqrt{7} - \sqrt{6} } - \frac{1}{ \sqrt{6} - \sqrt{5} } + \frac{1}{ \sqrt{5} - 2} = 5 }$
​

Answer 1

Answer:

Given √2

To prove: √2 is an irrational number.

Proof:

Let us assume that √2 is a rational number.

So it can be expressed in the form p/q where p, q are co-prime integers and q≠0

√2 = p/q

Here p and q are coprime numbers and q ≠ 0

Solving

√2 = p/q

On squaring both the side we get,

=>2 = (p/q)2

=> 2q2 = p2……………………………..(1)

p2/2 = q2

So 2 divides p and p is a multiple of 2.

⇒ p = 2m

⇒ p² = 4m² ………………………………..(2)

From equations (1) and (2), we get,

2q² = 4m²

⇒ q² = 2m²

⇒ q² is a multiple of 2

⇒ q is a multiple of 2

Hence, p, q have a common factor 2. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number

√2 is an irrational number

Answer 2

Answer:

LHS

13−8–√−18–√−7–√+17–√−6–√−16–√−5–√+15–√−2

3+8–√32−8–√2−8–√+7–√8–√2−7–√2+7–√+6–√7–√2−6–√2−6–√+5–√6–√2−5–√2+5–√+25–√−22

3+3–√−8–√−7–√+7–√+8–√−6–√−5–√+5–√+2

5=RHS.

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