Question

If p, q, n are three positive real numbers and p > q then which of the following is correct.

• A. $p^{\frac{-1}{n}}<q^{\frac{-1}{n}}$
• B. $p^{\frac{1}{n}}<q^{\frac{1}{n}}$
• C. $p^{\frac{-1}{n}}>q^{\frac{-1}{n}}$
• D. $p^n<q^n$

Answer 1

The correct option is A

$p^{\frac{-1}{n}}<q^{\frac{-1}{n}}$

Given that, p > q

$\Rightarrow \frac{1}{p}<\frac{1}{q}$ (using the property of inequality)
${\left(\frac{1}{p}\right)}^{\frac{1}{n}}<{\left(\frac{1}{q}\right)}^{\frac{1}{n}}$
Since $n$ is positive $\left(\frac{1}{n}\right)$ is also positive. So the inequality doesn't change when we raise both sides of the inequality with the positive number $\frac{1}{n}$.
Hence,
$p^{\frac{-1}{n}}<q^{\frac{-1}{n}}$
which is given by option (a).