Approximate square root of 53 to the nearest integer.
The Correct Answer and Explanation is :
The correct answer is: 7
To approximate the square root of 53 to the nearest integer, we begin by recognizing that the square root of a number ( n ) is the value ( x ) such that ( x^2 = n ). In this case, we are looking for ( \sqrt{53} ).
First, we need to identify two perfect squares between which 53 lies. The perfect square just below 53 is ( 49 ) (which is ( 7^2 )), and the perfect square just above it is ( 64 ) (which is ( 8^2 )). Thus, we can deduce:
[
7^2 < 53 < 8^2 \quad \Rightarrow \quad 7 < \sqrt{53} < 8
]
Next, we can refine our approximation by calculating ( 7.5^2 ) to see if it is closer to 53:
[
7.5^2 = 56.25
]
Since ( 56.25 ) is greater than ( 53 ), this tells us that ( \sqrt{53} ) is less than ( 7.5 ). To narrow it down further, we can try ( 7.2 ) and ( 7.3 ):
Calculating ( 7.2^2 ):
[
7.2^2 = 51.84
]
And calculating ( 7.3^2 ):
[
7.3^2 = 53.29
]
From these calculations, we find that:
[
51.84 < 53 < 53.29 \quad \Rightarrow \quad 7.2 < \sqrt{53} < 7.3
]
Since ( 7.2 ) is just below and ( 7.3 ) is just above 53, we can conclude that ( \sqrt{53} ) is approximately ( 7.28 ) when calculated more precisely. However, since we are looking for the nearest integer, we see that ( 7.28 ) rounds up to ( 7 ).
Thus, the square root of 53, rounded to the nearest integer, is:
[
\boxed{7}
]
This approximation demonstrates an essential mathematical skill: estimating square roots by comparing with known perfect squares and refining guesses through simple calculations.