MCQ
When simplified $\big(\text{x}^{-1}+\text{y}^{-1}\big)^{-1}$ is equal to:
- A$\text{xy}$
- B$\text{x}+\text{y}$
- ✓$\frac{\text{xy}}{\text{x}+\text{y}}$
- D$\frac{\text{x}+\text{y}}{\text{xy}}$
✓
Answer
Correct option: C.
$\frac{\text{xy}}{\text{x}+\text{y}}$
We have to simplify $\big(\text{x}^{-1}+\text{y}^{-1}\big)^{-1}$
So,
$\big(\text{x}^{-1}+\text{y}^{-1}\big)^{-1}=\Big(\frac{1}{\text{x}}+\frac{1}{\text{y}}\Big)^{-1}$
$=\frac{1}{\frac{1}{\text{x}}+\frac{1}{\text{y}}}$
$=\frac{1}{\frac{1\times\text{y}}{\text{x}\times\text{y}}+\frac{1\times\text{x}}{\text{y}\times\text{x}}}$
$=\frac{1}{\frac{\text{y}}{\text{xy}}+\frac{\text{x}}{\text{xy}}}$
$\big(\text{x}^{-1}+\text{y}^{-1}\big)^{-1}=\frac{1}{\frac{\text{y}+\text{x}}{\text{xy}}}$
$=\frac{\text{xy}}{\text{y}+\text{x}}$
The value of $\big(\text{x}^{-1}+\text{y}^{-1}\big)^{-1}$ is $\frac{\text{xy}}{\text{y}+\text{x}}$
Hence the correct choice is $c.$
So,
$\big(\text{x}^{-1}+\text{y}^{-1}\big)^{-1}=\Big(\frac{1}{\text{x}}+\frac{1}{\text{y}}\Big)^{-1}$
$=\frac{1}{\frac{1}{\text{x}}+\frac{1}{\text{y}}}$
$=\frac{1}{\frac{1\times\text{y}}{\text{x}\times\text{y}}+\frac{1\times\text{x}}{\text{y}\times\text{x}}}$
$=\frac{1}{\frac{\text{y}}{\text{xy}}+\frac{\text{x}}{\text{xy}}}$
$\big(\text{x}^{-1}+\text{y}^{-1}\big)^{-1}=\frac{1}{\frac{\text{y}+\text{x}}{\text{xy}}}$
$=\frac{\text{xy}}{\text{y}+\text{x}}$
The value of $\big(\text{x}^{-1}+\text{y}^{-1}\big)^{-1}$ is $\frac{\text{xy}}{\text{y}+\text{x}}$
Hence the correct choice is $c.$
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