Question

How do you simplify \[\dfrac{{{x^{ - 1}} - {y^{ - 1}}}}{{{x^2} - {y^2}}}\].

Answer 1

Hint: In order to solve these type of questions, use the properties of exponents such as \[{x^{ - a}} = \dfrac{1}{{{x^a}}}\], and the identities i.e.,\[{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)\], and the simplify numerator by taking L.C.M and taking out the negative sign and the expression will be simplified to \[x\] and \[y\] terms till we get the simplified term.

Complete step-by-step solution:
Given expression is \[\dfrac{{{x^{ - 1}} - {y^{ - 1}}}}{{{x^2} - {y^2}}}\],
This is expression is in \[x\] and \[y\], first we have to simplify the numerator using the exponents properties and simplify the denominator using \[{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)\], and then simplify the expression, till we get the simplifies term which cannot be simplified further.
Now using the properties of exponents such as \[{x^{ - a}} = \dfrac{1}{{{x^a}}}\], and the identities i.e.,\[{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)\],
Now applying the identities we get,
\[\dfrac{{{x^{ - 1}} - {y^{ - 1}}}}{{{x^2} - {y^2}}}\],
Now using the exponents identity i.e.,\[{x^{ - a}} = \dfrac{1}{{{x^a}}}\],and using the identity\[{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)\]in the denominator, we get,
\[ \Rightarrow \dfrac{{\dfrac{1}{x} - \dfrac{1}{y}}}{{\left( {x - y} \right)\left( {x + y} \right)}}\],
 Now taking LCM in the numerator and simplifying we get,
\[ \Rightarrow \dfrac{{\dfrac{{1 \times y}}{{x \times y}} - \dfrac{{1 \times x}}{{y \times x}}}}{{\left( {x - y} \right)\left( {x + y} \right)}}\],
Now multiplying the denominator in the numerator and simplifying we get,
\[ \Rightarrow \dfrac{{\dfrac{y}{{xy}} - \dfrac{x}{{xy}}}}{{\left( {x - y} \right)\left( {x + y} \right)}}\],
As the denominator is equal in the numerator,we get,
\[ \Rightarrow \dfrac{{\dfrac{{y - x}}{{xy}}}}{{\left( {x - y} \right)\left( {x + y} \right)}}\],
Taking the negative sign we get,
\[ \Rightarrow \dfrac{{\dfrac{{ - \left( {x - y} \right)}}{{xy}}}}{{\left( {x - y} \right)\left( {x + y} \right)}}\],
Now eliminating the like terms we get,
\[ \Rightarrow \dfrac{{\dfrac{{ - 1}}{{xy}}}}{{\left( {x + y} \right)}}\],
Now simplifying we get,
\[ \Rightarrow \dfrac{{ - 1}}{{xy\left( {x + y} \right)}}\].
The value of the given expression is \[\dfrac{{ - 1}}{{xy\left( {x + y} \right)}}\].

\[\therefore \]The value of the given expression is \[\dfrac{{ - 1}}{{xy\left( {x + y} \right)}}\], i.e.,\[\dfrac{{{x^{ - 1}} - {y^{ - 1}}}}{{{x^2} - {y^2}}} = \dfrac{{ - 1}}{{xy\left( {x + y} \right)}}\].

Note: In these types of questions, students should be careful in calculations and using the identities required, when to use these identities, and where to use the identities, and while taking LCM also students should be careful while multiplying and cancelling the terms.