Answer:
Explanation:
The product of the square root of 3 and the square root of 9 is rational. To understand why, we need to define what rational and irrational numbers are.
A rational number is any number that can be expressed as a fraction, where both the numerator and denominator are integers. An irrational number, on the other hand, cannot be expressed as a fraction and has an infinite non-repeating decimal representation.
In this case, the square root of 3 (√3) is an irrational number because it cannot be expressed as a fraction. However, the square root of 9 (√9) is equal to 3, which is a rational number since it can be expressed as the fraction 3/1.
When we multiply √3 by √9, we get:
√3 * √9 = √(3 * 9) = √27
Since 27 is not a perfect square, we can simplify √27 by factoring it into its prime factors:
√27 = √(3 * 3 * 3) = 3√3
Here, we can see that the product of √3 and √9 simplifies to 3√3. Although the expression contains an irrational component (√3), it is still considered rational because it can be written as a fraction (in this case, 3/1) where both the numerator and denominator are integers.
Therefore, the product of the square root of 3 and the square root of 9 is rational.