The Square root of 2

The square root of 2 is irrational. Just how do ns know? let me define ...

You are watching: Is the simplified form of 2square root of 3 − 2square root of 3 rational?

Squaring a reasonable Number

First, let us see what happens when we square a reasonable number:

If the rational number is a/b, then the becomes a2/b2 when squared.


Notice the the exponent is 2, i m sorry is one even number.

But to execute this effectively we should really rest the number down right into their prime factors (any whole number above 1 is element or have the right to be do by multiplying prime numbers):


Notice that the exponents room still also numbers. The 3 has an exponent that 2 (32) and the 2 has actually an exponent that 4 (24).

In some instances we may need to leveling the fraction:


Example: (1690)2

Firstly: 16 = 2×2×2×2 = 24, and 90 = 2×3×3×5 = 2×32×5

(1690)2 = (242×32×5)2

= (2332×5)2

= 2634×52


But one point becomes obvious: every exponent is an even number!

So we can see that as soon as we square a rational number, the an outcome is comprised of prime numbers whose exponents are all even numbers.


Back to 2

Now, let united state look at the number 2: can this have actually come about by squaring a rational number?


As a fraction, 2 is 2/1

Which is 21/11 ,and that has actually odd exponents!

Can we remove odd exponents?

We could write 1 as 12 (so it has an also exponent), and also then we have:

2 = 21/12

But over there is quiet an odd exponent (on the 2).

We have the right to simplify the whole thing to 21, however still an odd exponent.

We might even shot things prefer 2 = 4/2 = 22/21, yet we tho cannot get rid of an strange exponent

Oh no, there is always an odd exponent.

So it might not have been make by squaring a reasonable number!


This way that the value that to be squared to make 2 (ie the square source of 2) cannot be a rational number.

In various other words, the square root of 2 is irrational.

Try Some more Numbers


How around 3?

3 is 3/1 = 31

But the 3 has an exponent the 1, therefore 3 might not have been made by squaring a reasonable number, either.

The square root of 3 is irrational


How around 4?

4 is 4/1 = 22

Yes! The exponent is an even number! for this reason 4 have the right to be do by squaring a rational number.

The square source of 4 is rational


This idea can additionally be extended to cube roots, etc.

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Conclusion

To find if the square source of a number is irrational or not, examine to see if its prime components all have even exponents.

It likewise shows us there must be irrational number (such together the square source of two) ... In situation we ever before doubted it!