Question

which of the following is an irrational number? \\(\bigcirc\\) \\(\sqrt{196}\\) \\(\bigcirc\\) \\(\sqrt3{512}\\) \\(\bigcirc\\) \\(\sqrt5{243}\\) \\(\bigcirc\\) \\(\sqrt7{671}\\)

Explanation:

Step1: Analyze $\sqrt{196}$

A square root of a number is rational if the number is a perfect square. $14^2 = 196$, so $\sqrt{196}=14$, which is rational.

Step2: Analyze $\sqrt[3]{512}$

A cube root of a number is rational if the number is a perfect cube. $8^3 = 512$, so $\sqrt[3]{512}=8$, which is rational.

Step3: Analyze $\sqrt[5]{243}$

A fifth root of a number is rational if the number is a perfect fifth - power. $3^5=243$, so $\sqrt[5]{243} = 3$, which is rational.

Step4: Analyze $\sqrt[7]{671}$

We check if 671 is a perfect seventh - power. The seventh - power of integers: $2^7=128$, $3^7 = 2187$. Since 671 is not a perfect seventh - power, $\sqrt[7]{671}$ cannot be expressed as a fraction of two integers, so it is irrational.

Answer:

$\sqrt[7]{671}$ (the option corresponding to $\sqrt[7]{671}$)