8. is √42 rational or irrational? explain.

Answer:

Irrational

Explanation:

Step1: Recall the definition of rational and irrational numbers

A rational number can be written as $\frac{p}{q}$ where $p,q\in\mathbb{Z}$ and $q\neq0$. An irrational number cannot be written in this form.

Step2: Assume $\sqrt{42}$ is rational

Suppose $\sqrt{42}=\frac{p}{q}$ where $\gcd(p,q) = 1$ (lowest - terms). Then $42=\frac{p^{2}}{q^{2}}$, so $p^{2}=42q^{2}=2\times3\times7\times q^{2}$.

Step3: Analyze the prime - factorizations

Since $p^{2}$ has an even number of each prime factor in its prime - factorization, and the right - hand side $2\times3\times7\times q^{2}$ implies that the exponents of 2, 3, and 7 in the prime - factorization of $p^{2}$ are odd (because of the single factors 2, 3, and 7) unless $q = 0$ (which is not allowed) or there is a contradiction. So $\sqrt{42}$ cannot be written as a fraction $\frac{p}{q}$, and it is irrational.