Question

prove that 3 root 7 is an irrational number​

Answer 1

Given : 3√7

To Find : prove that  its irrational number

Solution:

Lets assume that  3√7 is not irrational number

Hence its a rational number

so  3√7 can be written as p/q  where p & q are co prime

3√7 = p/q

=> p  = 3√7q

Squaring both sides

=> p²  = 9 * 7 q²

As on left side its  a square and 7 is a prime number

Hence  q must of form 7n

q = 7n

=> p² = 9 * 7 (7n)²

=> p² = 9 * 7 * 7².n²

=> p must of form 7m

it means p & q must have a common factor 7

so o & q are not co prime

so our initial assumption that  3√7 is rational is wrong

Hence  3√7 is irrational

QED

Hence proved

Learn More:

Prove that 4-2√7 is an irrational number, given that 7 is an irrational ...

https://brainly.in/question/15078170

Prove that √p + √q is irrational, where p, q are primes. - Brainly.in

https://brainly.in/question/5481295

Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

TO PROVE

$\sf{ 3 \sqrt{7} \: \: \: is \: an \: irrational \: number}$

PROOF

First we will prove that √7 is an irrational number

Let us assume that √7 is a rational number.

Then as we know a rational number should

$\displaystyle \sf{be \: of \: the \: form \: \: \: \frac{p}{q} }$

where p and q are co- prime number.

$\displaystyle \sf{ \sqrt{7} = \frac{p}{q} \: \: \: \: where \: p \: and \: q \: are \: \: co- prime}$

√7q = p

Now by squaring both the sides

we get

(√7q)² = p²

7q² = p² ........ ( 1 )

So,

If 7 is the factor of p²

Then 7 is also a factor of p ......... ( 2 )

=> Let p = 7m , where m is any integer

Squaring both sides again

p² = (7m)²

p² = 49m²

Putting the value of p² in equation ( 1 )

7q² = p²

7q² = 49m²

q² = 7m²

So,

If 7 is factor of q²

Then 7 is also factor of q

Since

7 is factor of p & q both

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

Hence √7 is an irrational number

Here 3 is rational and √7 is irrational

Since the product of a rational number and an irrational number is an irrational number

Hence 3√7 is an irrational number

━━━━━━━━━━━━━━━━

LEARN MORE FROM BRAINLY

If x=( √3a+2b + √3a-2b) / ( √3a+2b - √3a-2b) prove that : bx²-3ax+b=0

https://brainly.in/question/19664646