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 $a$.
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 $a$.
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