kerrum75

2021-12-29

How do you find the principal square root of 8?

Jonathan Burroughs

For a non-negative real number, n the principal square root is the non-negative solution to ${x}^{2}=n$
The symbol $\sqrt{n}$ is used for the principal square root of n
The pricipal square root of 8, denoted $\sqrt{8}$ is the number whose sguare is 8. There is no easy method for finding this number. We use successive approximation (or other iterative techniques) to get increasingly accurate approximations.
We do, however write $\sqrt{8}$ in "simpler" form $2\sqrt{2}$
$\sqrt{8}=\sqrt{4\cdot 2}=\sqrt{4}\cdot \sqrt{2}=2\sqrt{2}$. Perhaps this is what you meant by "find"?

Travis Hicks

If, by "find" you mean get a decimal approximation, you could use: start with a number you know is close ${3}^{2}=9$ and ${2}^{2}=4$, so we'll start with 3 divide 8 by your last estimate: $8÷3=2.6667$ (to 4 decimal places)
Average your previous estimate and the quotient: $\frac{3+2.6667}{2}=2.8334$ (rounding)
Repeat:
Divide: $8÷2.8334=2.8235$
Average: $\frac{2.8334+2.8235}{2}=2.8284$
Repeat
Divide: $8÷2.8284=2.8284$
Average $=2.8284$
$\sqrt{8}\approx 2.8284$ this estimate is accurate to 4 decimal places.
If you need more accuracy, start again and keep more places when rounding.

nick1337