Question

Approximate square root 42 to the nearest tenth.
show me how too please?!!!

Answer 1

One way you can approximate square roots is via what's known as the Babylonian method.

Take a guess as to what your square root might be. We know that 6*6=36 and 7*7 = 49, so 6 is slightly closer. We'll use 6 as our guess.

Next take the number for which you are trying to find the square root, and divide it by the current guess. Add that number to the current guess, and then divide by 2. So our first iteration would be
\[(\frac{42}{6} + 6)/2 = (7+6)/2 = 6.5\]
6.5 is our new estimate. Now we do it again!
\[(\frac{42}{6.5} + 6.5)/2 = (6.46 + 6.5)/2 = 6.48\]
\[(\frac{42}{6.48} + 6.48)/2 = (6.48+6.48)/2 = 6.48\] and the value has stopped changing in the places we care about. Rounding to the nearest 0.1, that would be 6.5.

\[\sqrt{42} \approx 6.4807406984\]