Question
Prove that:
$\frac{\text{abc}}{\text{a}^{-1}\text{b}^{-1}+\text{b}^{-1}\text{c}^{-1}+\text{c}^{-1}\text{a}^{-1}}=\text{abc}$
$\frac{\text{abc}}{\text{a}^{-1}\text{b}^{-1}+\text{b}^{-1}\text{c}^{-1}+\text{c}^{-1}\text{a}^{-1}}=\text{abc}$
✓
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
In the given figure, O is the center of the circle. If $\angle\text{BOD}=160^\circ,$ find the value of x and y.
→Show that 1.272727 = $1. \overline {27}$ can be expressed in the form $\frac{p}{q}$, where p and q are integers and q $\ne$ 0.
→In a $\triangle\text{ABC},$ if $\angle\text{A}=55^\circ,\angle\text{B}=40^\circ,$ find $\angle\text{C}.$
→If two straight lines intersect each other, then prove that the ray opposite the bisector of one of the angles so formed bisects the vertically-opposite angle.
Find the values of x in each of the following:
→$2^{\text{x}-7}\times5^{\text{x}-4}=1250$
A joker's cap is in the form of a right circular cone of base radius 7cm and height 24cm. Find the area of the sheet required to make 10 such caps.
A hemispherical bowl is made of steel 0.5cm thick. The inside radius of the bowl is 4cm. Find the volume of steel used in making the bowl. $\big(\text{Take}\ \pi=\frac{22}{7}\big).$
→Simplify (2x - 5y)3 - (2x + 5y)3.
Find the following products:
(4x - 3y + 2z)(16x2 + 9y2 + 4z2 + 12xy + 6yz - 8zx)
(4x - 3y + 2z)(16x2 + 9y2 + 4z2 + 12xy + 6yz - 8zx)
Evaluate the following:
(103)3
(103)3