Question

prove that 2√5 is an irrational number also check whether (2√5+√7)(2√5-√7) is rational or irrational.
please answer this question as soon and possible and please don't give wrong answer or irrelevant answers.◉‿◉​

Answer 1

Answer:

a) Let us assume that 23+5 is rational number.

Let P=23+5 is rational

on squaring both sides we get

P2=(23+5)2=(23)2+(5)2+2×23×5

P2=12+5+415

P2=17+415

4P2−17=15   ………..(1)

Since P is rational no. therefore P2 is also rational & 4P2−17 is also rational.

But 15 is irrational & in equation(1)

4P2−17

Step-by-step explanation:

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Answer 2

Answer:

2√5= irrational

(2√5-√7)(2√5+√7) is rational.

Step-by-step explanation:

To prove :-

2√5= Irrational.

since,

We know 2 is a rational number so, Rational x irrational is always an irrational number.

So, proving √5 irrational can directly say that, 2√5 is irrational.

So, lets solve this with the method of contradiction.

______________________________

Let , √5 be a rational number, so this can be represented in the form of p/q

having no common multiples.

So, √5=p/q

squarring both sides.

=> (√5)²=(p/q)²

=> 5= p²/q²

=> 5q²=p²-----(1)

and,

=>p²/5= q²

Which means 5 divides p.

so, p is a multiple of 5.

therefore,

p=5m ... representing by "m".

squarring again.

=> p²=25m²--------(2)

So, from equation 1 and equation 2. We can say that

25m²=q²

which also have a common multiple of 5.

which is contradicting that, p/q has a common multiple.of 5, which avoids the rule of rational numbers , as stated above.

So, 2√5 is an irrational number.

.

___________________________

Now, to prove whether,

(2√5+√7)(2√5-√7) is a rational or irrational.

Since, this is the form of (a+b)(a-b)=a²-b²

So, directly using the identity here we go,

=> (2√5)²-(√7)²

=> 4x5-7

=>20-7

=>13.

Which is a rational number, thus its rational.

Ans.