## The Square root of 2

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

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### 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: (*16***90**)2

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

(*16***90**)2 = (*24***2×32×5**)2

= (*23***32×5**)2

= *26***34×52**

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 = 3**1**

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 = 2**2**

**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!