Is eight plus the square root of 32 greater than, equal to, or less than the square root of 40?
Let us consider how these two numbers can be compared without needing to use a calculator. Recall that if a given number 𝑎 is less than 𝑏 squared, where both 𝑎 and 𝑏 are positive, then the square root of 𝑎 is less than 𝑏. As an example of this, since eight is less than three squared, which is nine, then the square root of eight is less than three. This makes sense considering three is equal to root nine.
This property is very useful for estimating the value of square roots. For example, let us consider the square root of 40. If we consider square numbers that are around 40, then we can see that 40 is less than 49, which is seven squared, and greater than 36, which is six squared. So, this tells us that the value of the square root of 40 is somewhere between six and seven. However, let us now consider eight plus root 32. Since the principal square root of 32 must be a positive number, eight plus root 32 must be greater than eight.
Now, eight is certainly greater than seven. And we have already shown that root 40 is less than seven, since seven is root 49. Hence, we can conclude that eight plus the square root of 32 is greater than the square root of 40.