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The concept of the square root is fundamental in mathematics. It involves finding a number that, when multiplied by itself, yields the original number. Formally, if you have a number, say \( x \), and you want to find another number that when squared gives \( x \), you are looking for the square root of \( x \). For example, if you square 3 (meaning you multiply 3 by itself), you get 9. Consequently, the square root of 9 is 3, because \( 3^2 = 9 \).

In algebraic terms, if \( y^2 = x \), then \( y \) is the square root of \( x \). It's crucial to realize that every positive number actually has two roots: a positive and a negative one. This is because both a positive and a negative number, when squared, will result in a positive number. For instance, both \( 3^2 \) and \( (-3)^2 \) equal 9, so both 3 and -3 are square roots of 9.

In the context of 'principal square roots', we refer exclusively to the positive square root of a number. Taking the earlier example of the square root of 9, while both 3 and -3 are square roots of 9, the principal square root is 3, the positive one. This distinction is crucial because, in most mathematical contexts, when the term 'square root' is used without a sign in front of the root symbol, it implies the principal, or positive, square root.

Returning to the original exercise, the principal square root of a positive number is 'the positive number'. Understanding this helps to readily identify the principal square root in any given situation and avoid confusion with its negative counterpart.

Square roots have several important properties that help in solving mathematical problems. Firstly, the square root of a product of two numbers is equal to the product of the square roots of each number — that is, \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). However, this rule only applies if both \( a \) and \( b \) are non-negative. Another property is that the square root of a quotient is the quotient of the square roots of the numerator and the denominator, \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \), provided that \( b \) is not zero.

Square roots also obey distributive properties similar to other algebraic functions. It is important to remember that while the square root of a sum is not the sum of the square roots, \( \sqrt{a + b} eq \sqrt{a} + \sqrt{b} \), a common mistake. Keeping these properties in mind can greatly assist in simplifying and solving equations involving square roots.