What is the square root of 9? The Square Root of 9 is an irrational number, which means that its decimal representation will never end or repeat itself.
It can be expressed as a fraction with an infinite number of digits behind the decimal point.
We are going to use algebraic notation and the fundamental theorem of arithmetic to prove this statement true. Be sure you have a piece of paper handy because there will be some writing involved! Let’s get started…
The square root of 9, 3, can be simplified as the fraction 3/1. Remember that any even or odd number can be written as a fraction with an infinite number of digits behind the decimal point. The only difference between these two fractions is where we place the 1 in relation to the 3. We know that the square root of 9 can be simplified as a fraction with an infinite number of digits behind the decimal point, however, the fundamental theorem of arithmetic tells us that all positive integers have at least one pair of unique prime factors. For example, 1 = 1×1, 2 = 2×1, 3 = 3×1
√9 = ±3 can be written in the form of a fraction 3/1. It proves that √9 is a rational number.
the principal square root of 9 is sqrt(9) = +3
9 is an odd number as it can be written as 9 = 3 × 3. It’s also a prime number, because its only factors are 1 and 9.
The negative square root of 9 is – √9, so it is a rational number
The square root of any odd number except 3 is an irrational number
because ∛9 = ∛(3 × 3). It cannot be expressed in the form of p/q. Therefore, the value of the cube root of 9 is an irrational number.
We can prove that both 3 and – √9 are rational numbers. The other two roots of 9, ∛3 and ∛-(3 × 3) are irrational numbers. That is because the number 9 has no prime factors to simplify it as a fraction with an infinite number of digits behind the decimal point.