Find the two whole numbers that are the closest to √42.
The Correct Answer and Explanation is:
To find the two whole numbers that are closest to ( \sqrt{42} ), let’s first estimate the square root of 42 and then determine which whole numbers are nearest.
Step 1: Understanding square roots
The square root of a number ( n ) is the number that, when multiplied by itself, gives ( n ). In mathematical terms:
[
\sqrt{n} = x \quad \text{where} \quad x^2 = n.
]
We are asked to find the square root of 42, so we need to determine which two whole numbers ( x ) and ( x+1 ) (or ( x-1 )) are closest to ( \sqrt{42} ).
Step 2: Estimating ( \sqrt{42} )
We know that ( 6^2 = 36 ) and ( 7^2 = 49 ). Since ( 42 ) is between these two perfect squares, ( \sqrt{42} ) must be between 6 and 7.
To be more precise, we can estimate:
- ( 6.5^2 = 6.5 \times 6.5 = 42.25 )
- ( 6.4^2 = 6.4 \times 6.4 = 40.96 )
The square of 6.5 is 42.25, which is very close to 42, and the square of 6.4 is 40.96, which is a little lower. Therefore, we can conclude that ( \sqrt{42} ) is slightly less than 6.5.
Step 3: Closest whole numbers
Since ( \sqrt{42} ) is between 6 and 7, and it is closer to 6.5 than to 7, we can say that the two whole numbers closest to ( \sqrt{42} ) are 6 and 7.
Final Answer:
The two whole numbers closest to ( \sqrt{42} ) are 6 and 7.
This estimation technique helps us find the closest whole numbers without using a calculator. The process involves comparing the square of different numbers to find the range in which the square root lies.