Question

How do you use the square roots of $144$ and $169$ to estimate the square root of $154$?

Answer 1

Hint: In the given question, we are required to estimate the value of the square root of a given number using the square roots of perfect squares of numbers that are close to the given number. So, we need thorough knowledge of squares and square roots. We also need to remember the squares of the first twenty natural numbers so as to get a clear idea about the square root of $154$ and to estimate its value correctly.

Complete step-by-step answer:
We have to estimate the value of the square root of $154$ using the square roots of $144$ and $169$. So, we first have to know the value of square roots of $144$ and $169$. So, square root of $144$ is $12$ and square root of $169$ is $13$. These square roots can easily be computed using the prime factorization method.
Now, we can observe that the value $154$ is closer to $144$ than to $169$. Hence, the square root of $154$ will be closer to $12$ than to $13$. We also know that the square root of $154$ is less than $12.5$ as square of $12.5$ is $156.25$. Hence, We can predict that the value of the square root of $154$ will be somewhere around $12.4$.

Note: Prime factorization method can also be employed to find the prime factors of the given number and then taking out the numbers from the square root radical sign that exist in pairs. The square root of $154$ can be calculated using the long division method that is the standard method to calculate square root of any particular number.