Question
Prove that: $\frac{\text{abc}}{\text{a}^{-1}\text{b}^{-1}+\text{b}^{-1}\text{c}^{-1}+\text{c}^{-1}\text{a}^{-1}}=\text{abc}$
✓
Answer
$\frac{\text{abc}}{\text{a}^{-1}\text{b}^{-1}+\text{b}^{-1}\text{c}^{-1}+\text{c}^{-1}\text{a}^{-1}}=\text{abc}$
Left hand side $(LHS)$ = Right hand side $(RHS)$
Considering $LHS$, $=\frac{\text{a}+\text{b}+\text{c}}{\frac{1}{\text{ab}}+\frac{1}{\text{bc}}+\frac{1}{\text{ca}}}$
$=\frac{\frac{\text{a}+\text{b}+\text{c}}{\text{a}+\text{b}+\text{c}}}{\text{abc}}$
$=\text{abc}$ Therefore, $LHS = RHS$ Hence proved
Left hand side $(LHS)$ = Right hand side $(RHS)$
Considering $LHS$, $=\frac{\text{a}+\text{b}+\text{c}}{\frac{1}{\text{ab}}+\frac{1}{\text{bc}}+\frac{1}{\text{ca}}}$
$=\frac{\frac{\text{a}+\text{b}+\text{c}}{\text{a}+\text{b}+\text{c}}}{\text{abc}}$
$=\text{abc}$ Therefore, $LHS = RHS$ Hence proved
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
If $a-b=4$ and $a b=21$, find the value of $a^3-b^3$
→If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
Represent geometrically the following numbers on the number line: $\sqrt{8.1}$
→Find the value of $x$ for which the angles $(2x - 5)^\circ$ and $(x - 10)^\circ $ are the complementary angle.
→$\ce{\triangle ABC}$ is an isosceles triangle in which $\ce{AB = AC}$. Side $BA$ is produced to $D$ such that $\ce{AD = AB}$. Show that $\ce{\angle BCD}$ is a right angle.
→Factorize the following expressions:
$8 a^3-b^3-4 a x+2 b x$
→$8 a^3-b^3-4 a x+2 b x$
In a $\triangle\text{ ABC},\text{ AD}$ bisects $\angle\text{A}$ and $\angle\text{C} > \angle\text{B}.$. Prove that $\angle\text{ADB} > \angle\text{ADC}.$
→In Figure, LM is a line parallel to the y-axis at a distance of 3 units.
i. What are the coordinates of the points P, R and Q?
ii. What is the difference between the abscissa of the points L and M?
i. What are the coordinates of the points P, R and Q?
ii. What is the difference between the abscissa of the points L and M?
Verify that $x^3+y^3+z^3-3 x y z=\frac{1}{2}(x+y+z)\left[(x-y)^2+(y-z)^2+(z-x)^2\right]$.
→If $2^\text{x}\times3^\text{y}\times5^\text{z}=2160,$ find $x , y $and $z$. Hence, compute the value of $3^\text{x}\times2^{-\text{y}}\times5^{-\text{z}}.$
→