Question

show that there is no rational number whose square is 3.​

Answer 1

Step-by-step explanation:

if √3 is irrational then there exists no rational number with square 3

to prove :√3 is irrational

proof:Let us assume to the contrary that √3 is a rational number.

It can be expressed in the form of p/q

where p and q are co-primes and q≠ 0.

⇒ √3 = p/q

⇒ 3 = p2/q2 (Squaring on both the sides)

⇒ 3q2 = p2………………………………..(1)

It means that 3 divides p2 and also 3 divides p because each factor should appear two times for the square to exist.

So we have p = 3r

where r is some integer.

⇒ p2 = 9r2………………………………..(2)

from equation (1) and (2)

⇒ 3q2 = 9r2

⇒ q2 = 3r2

Where q2 is multiply of 3 and also q is multiple of 3.

Then p, q have a common factor of 3. This runs contrary to their being co-primes. Consequently, p / q is not a rational number. This demonstrates that √3 is an irrational number.

THANK ME AND MARK AS BRAINLIEST PLZZZZZ

Answer 2

SOLUTION

TO PROVE

There is no rational number whose square is 3.

EVALUATION

We have to show that there is no rational number whose square is 3.

If possible let there is a rational number x whose square is 3.

Then x² = 3

⇒ x = √3

Since x is a a rational number , then it should be in the form of p/q

where p and q are co- prime number.

So,

√3 = p/q { where p and q are co- prime}

⇒ √3q = p

Now, by squaring both the side

we get,

⇒ (√3q)² = p²

⇒ 3q² = p² - - - - - - - - (1)

So

If 3 is the factor of p²

then, 3 is also a factor of p - - - - - - - (2)

⇒ Let p = 3m { where m is any integer }

Squaring both sides

p² = (3m)²

⇒ p² = 9m²

putting the value of p² in Equation 1

3q² = p²

⇒ 3q² = 9m²

⇒ q² = 3m²

So,

if 3 is factor of q²

then, 3 is also factor of q

Since

3 is factor of p & q both

So, our assumption that p & q are co- prime is wrong

Therefore √3 is an irrational number

Hence there is no rational number whose square is 3.

Hence the proof follows

━━━━━━━━━━━━━━━━

Learn more from Brainly :-

$1.$ Write 18.777. . . in p/q form.

https://brainly.in/question/29915810

2. is the following is a rational number between 4 and 8

https://brainly.in/question/28372112

3. sum of rational numbers whose absolute value is 7/3

https://brainly.in/question/30899277