What is the square root of 181?

Understand the Problem

The question is asking for the square root of the number 181, which is a mathematical calculation to determine the value that, when multiplied by itself, gives 181.

Answer

13.453

Steps to Solve

Identify the type of number 181 is (prime or non-prime)

In order to compute the square root, it's helpful to know if 181 is a prime number which might influence the accuracy of our estimation methods. However, whether it is prime or not, we need a numerical method to find its square root since 181 is not a perfect square.

Estimate the square root using nearby perfect squares

We know that: (13^2=169) and (14^2=196)

Since (169 < 181 < 196), the square root of 181 lies between 13 and 14.

Use the average method for a better estimate

Find a more precise estimate by taking the average of 13 and 14 and refining it: $$ ext{Average} = \frac{13 + 14}{2} = 13.5$$ $$13.5^2 = 182.25$$

Since (182.25 > 181), the square root is slightly less than 13.5.

Refine the estimate using the average of 13.4 and 13.5

Calculate next estimate: $$ ext{Average} = \frac{13.4 + 13.5}{2} = 13.45$$ $$13.45^2 = 180.60$$

Since (180.60 < 181), the square root is slightly more than 13.45.

Further refine the estimate within 13.45 to 13.5

Take the average of 13.45 and 13.5: $$ ext{Average} = \frac{13.45 + 13.5}{2} = 13.475$$ $$13.475^2 = 181.427625$$

Since (181.427625 > 181), our square root lies slightly less than 13.475.

Final refinement within 13.45 to 13.475

By taking further averages and testing, we can pin down to a more accurate value, typically needing a calculator. By examination, it comes out to be approximately 13.453.

The square root of 181 is approximately 13.453

More Information

The square root of 181 is an irrational number, meaning it cannot be precisely represented as a simple fraction.

Tips

A common mistake is to stop the estimation too early without enough refinement, leading to a less accurate answer.