The square root of twenty-two to the nearest ten-thousandths.
- with a calculator- would you get an answer of about 4.7??

Calculator?

Yes, your answer is close, but not to the ten-thousandth.

Are you supposed to do this by hand? It's easy with a calculator.

Try the Babylonian method. It works great for doing square roots by hand.

a_{n+1}=\frac{1}{2}(a+\frac{b}{a})

This is a recursion, where k= an estimate and b is what your finding the squre root of.

Here's an example:

Suppose you want to find \sqrt{11}

We know it's going to be between 3 and 4. Because 3^3=9 and 4^2=16.

Make an intitial guess of, say, 3.3.

\frac{1}{2}(3.3+\frac{11}{3.3})=3.3166666...

\frac{1}{2}(3.31666+\frac{11}{3.31666})=3.31662479 062

Keep that up until you get the desired accuracy. But, I believe we're there already.

Now do it with \sqrt{22}.