If a, b, c are positive real numbers, then a^ -1 b b^ -1 c c^ -1 a is equal to

If a, b, c are positive real numbers, then a^ -1 b b^ -1 c c^ -1 …

MCQ

If $a, b, c$ are positive real numbers, then $\sqrt{\text{a}^{-1}\text{b}}\times\sqrt{\text{b}^{-1}\text{c}}\times\sqrt{\text{c}^{-1}\text{a}}$ is equal to

✓
$1$
B
$\text{abc}$
C
$\sqrt{\text{abc}}$
D
$\frac{1}{\text{abc}}$

✓

Answer

Correct option: A.

$1$

We have to find the value of $\sqrt{\text{a}^{-1}\text{b}}\times\sqrt{\text{b}^{-1}\text{c}}\times\sqrt{\text{c}^{-1}\text{a}}$ when $a, b, c$ are positive real numbers.
So,
$\sqrt{\text{a}^{-1}\text{b}}\times\sqrt{\text{b}^{-1}\text{c}}\times\sqrt{\text{c}^{-1}\text{a}}$
$=\sqrt{\frac{1}{\text{a}}\times\text{b}}\times\sqrt{\frac{1}{\text{b}}\times\text{c}}\times\sqrt{\frac{1}{\text{c}}\times}\text{a}$
$=\sqrt{\frac{\text{b}}{\text{a}}}\times\sqrt{\frac{\text{c}}{\text{b}}}\times\sqrt{\frac{\text{a}}{\text{c}}}$
Taking square root as common we get
$\sqrt{\text{a}^{-1}\text{b}}\times\sqrt{\text{b}^{-1}\text{c}}\times\sqrt{\text{c}^{-1}\text{a}}=\sqrt{\frac{\text{b}}{\text{a}}\times\frac{\text{c}}{\text{b}}\times\frac{\text{a}}{\text{c}}}$
$\sqrt{\text{a}^{-1}\text{b}}\times\sqrt{\text{b}^{-1}\text{c}}\times\sqrt{\text{c}^{-1}\text{a}}=1$
Hence the correct alternative is $a.$