square root of 213
Understand the Problem
The question is asking for the square root of 213, which is a basic mathematical operation that involves finding a number that, when multiplied by itself, equals 213.
Answer
The square root of 213 is approximately $14.5945$.
Answer for screen readers
The square root of 213 is approximately $14.5945$.
Steps to Solve
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Identify the problem We need to find the square root of 213, expressed as $\sqrt{213}$.
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Estimate the square root To estimate the square root, we identify the two perfect squares between which 213 falls. We know that $14^2 = 196$ and $15^2 = 225$. Therefore, we can estimate that:
$$ 14 < \sqrt{213} < 15 $$
- Refine the estimate To refine our estimate, we can try finding decimal approximations. Let’s check $14.5$:
$$ 14.5^2 = 210.25 $$
Since $210.25 < 213$, we can say:
$$ 14.5 < \sqrt{213} $$
Now, let's check $14.6$:
$$ 14.6^2 = 213.16 $$
Since $213.16 > 213$, we conclude:
$$ 14.5 < \sqrt{213} < 14.6 $$
- Use a calculator for accuracy To get a more precise value, using a calculator will give us:
$$ \sqrt{213} \approx 14.5945 $$
The square root of 213 is approximately $14.5945$.
More Information
The square root of 213 is an irrational number and cannot be simplified to a fraction, meaning it has non-repeating, non-terminating decimal places. It's also interesting that square roots often appear in various fields, such as physics and engineering.
Tips
- Rounding too early: Avoid rounding numbers too soon in the calculation process as it can lead to inaccuracies in the final answer.
- Skipping checking perfect squares: It’s important to establish bounds first using perfect squares to help with estimating the square root.
