Question

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

• A. (a) $p^{\left(\frac{-1}{n}\right)}<q^{\frac{-1}{n}}$
• B. (b) $p^{\frac{1}{n}}<q^{\frac{1}{n}}$
• C. (c) $p^{\frac{-1}{n}}>q^{\frac{-1}{n}}$
• D. (d) $p^n<q^n$

Answer 1

The correct option is A (a) $p^{\left(\frac{-1}{n}\right)}<q^{\frac{-1}{n}}$

Given that, p > q

⇒ $\frac{1}{p}<\frac{1}{q}$ (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).