Question

How do I find this out

Answer 1

If you take the square root of 3, you get ~1.732, which is between 1.7 and 1.8

Answer 2

Take some guesses and try them out.

Whole numbers:

Is the lower number 2 ?
If it is, then 2² is less than 3.
2² is 4.  That's more than 3.  So 2 is too big to be the lower number.
2 can be the upper number.

Is the lower number 1 ?
If it is, then 1² is less than 3.
1² is 1.  1 is less than 3.
So 1 can be the lower number.

√3  is between the two rational numbers 2 and 3.
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Numbers with 1 decimal place:

Can the lower number be 1.9 ?
If it can, then 1.9² is less than 3.
1.9² is 3.61.  That's more than 3. 
So 1.9 is too big to be the lower number.

Can the lower number be 1.8 ?
If it can, then 1.8² is less than 3.
1.8² is 3.24.  That's more than 3. 
So 1.8 is too big to be the lower number.

Can the lower number be 1.7 ?
If it can, then 1.7² is less than 3.
1.7² is 2.89.  Yay !  That's less than 3. 
So 1.7 can be the lower number.

√3  is between the rational numbers  1.7  and  1.8 .
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Numbers with 3 decimal places:

Can the lower number be 1.741 ?
If it can, then 1.741² is less than 3.
1.741² is 3.031... .  That's more than 3. 
So 1.741 is too big to be the lower number.

Can the lower number be 1.735 ?
If it can, then 1.735² is less than 3.
1.735² is 3.0102... .  That's more than 3. 
So 1.735 is too big to be the lower number.

Can the lower number be 1.734 ?
If it can, then 1.734² is less than 3.
1.734² is 3.0067... .  That's more than 3. 
So 1.734 is too big to be the lower number.

Can the lower number be 1.733 ?
If it can, then 1.733² is less than 3.
1.733² is 3.00328... .  That's more than 3. 
So 1.733 is too big to be the lower number.

Can the lower number be 1.732 ?
If it can, then 1.732² is less than 3.
1.732² is 2.9998... .  Yay!  That's less than 3. 

√3  is between the rational numbers  1.732  and 1.733 .
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Of course, if you already KNOW the square root of 3,
or if you go to a book or a calculator and find out what
it is, or if you know how to figure out what it is and you
take a pencil and paper and you do that, then you can
just pick any old rational number that's bigger and any
old rational number that's smaller, and say "THERE ! 
Here are two rational numbers that  √3 is between."

On my calculator,  √3 starts out with 

                                         1.7320508 ...
So it must be between
the two rational numbers     1.7320507
                     and              1.7320509 .
____________________________

In the question you posted, they GAVE you some pairs
of rational numbers.  You can test each pair and see
if it works.
If the √3 is between them, then
(the smaller one)² must be less than 3, and
(the bigger one)² must be more than 3.

Let's checkum out:

A).  1.2 and 1.3
     (1.2)² = 1.44 .  Could be !  Let's check the other one.
     (1.3)² = 1.69 .  Aw shucks. 
                           The √3 is bigger than both of them, not between them.

B).  1.7 and 1.8
      (1.7)² = 2.89 .  Could be!  Let's check the other one.
      (1.8)² = 3.24.   Yay !  The √3  is between these.
    
C).  2.6 and 2.7
     (2.6)² = 6.76.  Phooey!  The √3 can't be 'between' these, because
                           it's even smaller than the smaller number.

D).  2.7 and 2.8
     (2.7)² = 7.29.  Phooey!  The √3 can't be 'between' these, because
                           it's even smaller than the smaller number.