**Introduction**

Happy New
Year to our readers! I wish this year
will be a fruitful one for everybody.

Today, I
will illustrate how to calculate square roots by hand, using 54 756 as
an example. It is similar to long
division, but has some modifications.

**Solution**

Starting
from the right, pair up the digits.

2×2 =
4 is the nearest perfect square to 5. Subtract
and bring down the next two digits, giving 147.

Double the digit
2
to get 4. Think: ? × 4?
gives 147 or nearest possible value. We have 3×43 = 129.

Subtracting
and bringing down the next two digits gives
1856. Replicate the digit 4 on
the left and double the digit 3, giving
46.

Now think: ? × 46?
gives 1856 or nearest possible value. It turns out that 4 × 464 gives exactly 1856.
We are done! The square root
of 54 756 is
234.

**How does it work?**

This relies on the algebraic identity (10

*a*+

*b*)² = 100

*a*² + 20

*ab*+

*b*², the right-hand expression is equal to 100

*a*² + (20

*a*+

*b*)

*b*. For example, at stage 4, we have

*a*= 23,

*b*= 4 and (20

*a*+

*b*) = 464.

Did you
learn something today?