Question

If a, b, c are real numbers, then the value of $\sqrt{a^{-1}b}.\sqrt{b^{-1}c}.\sqrt{c^{-1}a}$ is

• A. abc
• B. $\sqrt{abc}$
• C. $\frac{1}{abc}$
• D. 1

Answer 1

The correct option is D 1

$\sqrt{a^{-1}b}\cdot \sqrt{b^{-1}c}\cdot \sqrt{c^{-1}a}$

$={\left(a^{-1}b\right)}^{\frac{1}{2}}\cdot {\left(b^{-1}c\right)}^{\frac{1}{2}}\cdot {\left(c^{-1}a\right)}^{\frac{1}{2}}$

$=a^{-\frac{1}{2}}{b}^{\frac{1}{2}}\cdot b^{-\frac{1}{2}}{c}^{\frac{1}{2}}\cdot c^{-\frac{1}{2}}{a}^{\frac{1}{2}}$

$=a^{\frac{1}{2}+\frac{1}{2}}\cdot b^{\frac{1}{2}-\frac{1}{2}}\cdot c^{\frac{1}{2}-\frac{1}{2}}$

$=a^0\cdot b^0\cdot c^0$

$=1\times 1\times 1=1$