Find the length of the hypotenuse for a right triangle with sides of 4
and 6 as irrational number.

The length of the hypotenuse is ~7.21

C= sq root (a^2+ b^2)
= sq root (4^2 + 6^2)
= sq root (16+36)
= sq root (52)

To find the square root of 52 without a calculator,
As you know 7^2 = 49 and 8^2 = 64 so you know the answer is somewhere between the two.
I computed 7.5^2 and I got 56.25 (too large)
Then I computed 7.25^2 and I got 52.26 (small but getting closer)
I computed 7.24^2 which equals 52.41 ( a little bit closer)
I computed 7.23^2 which equals 52.27 (a little closer)
I computed 7.22^2 which equals 52.13 (closer)
I computed 7.21^2 which equals 51.98 close enough!