True or False square root of 41 is a rational number. Explain

To determine whether the square root of 41 is a rational number, we need to understand the definition of a rational number.

A rational number is a number that can be expressed as a fraction, where the numerator and denominator are both integers and the denominator is not zero.

In the case of the square root of 41, we need to determine whether it can be expressed as a fraction in this form.

If we assume that the square root of 41 is a rational number (let's call it x), we can express it as x = a/b, where a and b are integers with b ≠ 0.

Squaring both sides of the equation, we get x^2 = (a^2)/(b^2).

Replacing x with the square root of 41, we obtain (√41)^2 = (a^2)/(b^2), which simplifies to 41 = (a^2)/(b^2).

Now, let's consider the prime factorization of 41: 41 is a prime number and cannot be expressed as the product of two smaller integers.

If we assume that a and b have a common factor other than 1, we could divide both sides of the equation by the common factor to simplify it further. However, since 41 is prime, a and b do not have any common factors other than 1.

Hence, the only way for (a^2)/(b^2) to be equal to 41 is if both a^2 and b^2 are equal to 41.

In other words, a and b themselves would have to be the square roots of 41.

However, the square root of 41 is an irrational number, meaning it cannot be expressed as a fraction. This contradicts our assumption that the square root of 41 is rational.

Therefore, we can conclude that the square root of 41 is not a rational number.