What is sqrt(48) ?

2 Answers
Apr 28, 2017

4sqrt3

Explanation:

Like Asma said you can calculate the square root on a calculator for the exact amount. But sometimes a professor will ask you to simplify. This will be very useful in the future.

sqrt48

We want to check if there is a number with a perfect square that is a multiple of the number in our square root. In this case, there is one.
16*3=48

Since 16 has a perfect square root of 4 we can solve this like this...

sqrt48

sqrt(16*3) <--- [Move square root of 16 outside of the root]

4sqrt3

May 13, 2017

sqrt(48) = 4sqrt(3) ~~ 18817/2716 ~~ 6.92820324

Explanation:

Note that:

48 = 4^2*3

So we find:

sqrt(48) = sqrt(4^2*3) = sqrt(4^2)sqrt(3) = 4sqrt(3)

That is the "simplest" form.

sqrt(48) is an irrational number a little less than 7, since 7^2 = 49.

In fact, since 48=7^2-1 is in the form n^2-1 its square root can be expressed as a continued fraction with a simple pattern:

sqrt(48) = [6;bar(1,12)] = 6+1/(1+1/(12+1/(1+1/(12+1/(1+1/(12+..))))))

To get rational approximations, we can truncate this continued fraction early (preferably just before a "12")...

For example:

sqrt(48) ~~ 6+1/(1+1/(12+1/1)) = 97/14 = 6.9bar(285714)

sqrt(48) ~~ 6+1/(1+1/(12+1/(1+1/(12+1/1)))) = 1351/195 = 6.9bar(282051)

sqrt(48) ~~ 6+1/(1+1/(12+1/(1+1/(12+1/(1+1/(12+1/1)))))) = 18817/2716 ~~ 6.92820324

A calculator tells me that it is closer to 6.92820323, but that's not bad.