Find all real square roots of 36
The correct answer and explanation is:
The real square roots of 36 are 6 and -6.
To explain this, recall that a square root of a number is a value that, when multiplied by itself, gives the original number. For 36, we need to find the number xx such that: x2=36x^2 = 36
There are two possible solutions because both positive and negative numbers, when squared, yield the same result. Specifically: 62=366^2 = 36 (−6)2=36(-6)^2 = 36
This is due to the fact that multiplying two positive numbers or two negative numbers results in a positive product. In other words, the square of both 6 and -6 is 36. Therefore, the square roots of 36 are both 6 and -6.
However, typically when referring to the square root symbol 36\sqrt{36}, it is understood to mean the positive square root, which is 6. If the problem asks for all real square roots, both 6 and -6 should be considered as valid answers.
This dual nature of square roots arises from the fact that the square function x2x^2 is not one-to-one: both positive and negative values of xx give the same output when squared. To differentiate, mathematicians use the principal square root, denoted as 36\sqrt{36}, to refer to the positive root only.
In summary, while the principal square root of 36 is 6, the complete set of real square roots includes both 6 and -6.