As a domain expert in mathematics, I would like to delve into the concept of rational numbers and their relationship with square roots, specifically focusing on the square root of 9.

A rational number is a term that refers to any number that can be expressed as the quotient (or ratio) of two integers, where the denominator is not zero. This definition encompasses a wide range of numbers, including all integers, fractions, and even some decimal numbers that have a repeating pattern. For instance, the number 4 can be expressed as \( \frac{4}{1} \), the fraction \( \frac{1}{2} \) is a rational number as well, and the repeating decimal 0.333... (which is equivalent to \( \frac{1}{3} \) ) is also rational because it can be expressed as a fraction of two integers.

Now, let's consider the concept of a square root. The square root of a number \( a \) is a value that, when multiplied by itself, gives the original number \( a \). It is denoted as \( \sqrt{a} \). For example, the square root of 4 is 2, because \( 2 \times 2 = 4 \). However, it's important to note that every positive real number has two square roots: one positive and one negative. For instance, the square roots of 9 are \( +3 \) and \( -3 \), since \( 3 \times 3 = 9 \) and \( (-3) \times (-3) = 9 \). The positive root is often referred to as the principal square root.

The question at hand is whether the square root of 9, which is 3, is a rational number. To determine this, we must see if it can be expressed as the quotient of two integers. The number 3 is an integer itself, and it can be expressed in the form \( \frac{3}{1} \), where both the numerator and the denominator are integers, and the denominator is not zero. Therefore, by the definition of a rational number, the square root of 9, which is 3, is indeed a rational number.

It's also worth mentioning that not all square roots are rational. For example, the square root of 2 is known to be an irrational number because it cannot be expressed as a simple fraction of two integers. Its decimal expansion is non-repeating and infinite.

In conclusion, the square root of 9, being 3, is a rational number because it meets the criteria of being expressible as the quotient of two integers with a non-zero denominator. This aligns with the provided information that "The square root of 9 is 3, since 3 * 3 = 9."

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3 is the principal square root of 9; -3 is also a square root of 9, but it is not principal square root. rational numberA rational number is a number that can be expressed as the quotient of two integers, with the denominator not equal to zero. ... The square root of 9 is 3, since 3 * 3 = 9.May 31, 2017read more >>