What is the square root of 214?

Steps to Solve

1. Identify the Square Root Operation

To find the square root of a number, we are looking for a value that, when multiplied by itself, gives us that number. In this case, we need to find $\sqrt{214}$.

2. Estimate the Square Root

We can find two perfect squares around 214 to make an initial estimate.

• The square of 14 is $14^2 = 196$.
• The square of 15 is $15^2 = 225$.

So, we know that $\sqrt{214}$ is between 14 and 15.

3. Refine the Estimate Using Averaging

Using a method of averaging can help to narrow it down further.

• First, let's find the average of 14 and 15: $$ \text{Average} = \frac{14 + 15}{2} = 14.5 $$

Next, we square this average: $$ 14.5^2 = 210.25 $$

Since $210.25 < 214$, we know $\sqrt{214}$ is greater than 14.5.

4. Try a Larger Value

Now, let's try a number slightly larger than 14.5, such as 14.6: $$ 14.6^2 = 213.16 $$

Since $213.16 < 214$, we try a slightly larger number, like 14.7: $$ 14.7^2 = 216.09 $$

Now we see that $14.6 < \sqrt{214} < 14.7$.

5. Final Refinement

Further refining, we can try 14.64: $$ 14.64^2 = 214.0996 $$

And 14.63: $$ 14.63^2 = 214.0569 $$

6. Result

Using a calculator or further iterative methods, we can find that: $$ \sqrt{214} \approx 14.628 $$