Square Roots

Links and useful resources

Inverse operations
squares and square roots
$\sqrt{x}$ is the positive value by definition, even though there is a matching negative!
We're not going to take the square root of any negative numbers for now, because that would just be unimagineable...
Since the square root like this is defined to be the positive number, we have to explicitly state that we want both positive and negative when that's the case. We do that by writing it with the $\pm$ symbol (\pm in latex): $x = \pm \sqrt{17}$ for example

When we're calculating square roots without a calculator, it really pays to get good at estimating.

Example: Simplify $\sqrt{65536}$

First, notice that this is close to 64000, which we can break into $64 \times 1000$
$64 = 8^{2}$ , and $1000 \approx 33^{2}$ , so we can say $\sqrt{65536} \approx 8 \times 33$
Now divide by $8 \times 33 = 264$ : $65536 \div 264 \approx 248$ , so our number is a bit big
We can zero in by picking the middle value and trying again: $\frac{264 + 248}{2} = 256$
In this case, we got the answer! $65536 = 256^{2}$

The key steps:

Estimate the approximate size of the answer by ignoring the small place-values and picking a square that's close to what you're looking for
Look at the digits: the ones' place will tell you what the possible answer digits are (if it's a perfect square - if not, you will know because there's no square with the right ones' place).
Divide by your guess, and use the difference to zero in on the answer. A good starting point is to pick the average of the two numbers for each step.