Question

( x to the power minus 2 minus y to the power minus 2 ) ➗ ( x to the power minus 1 plus y to the power minus 1)

Answer 1

To simplify the expression \(( x^{-2} - y^{-2} ) \div ( x^{-1} + y^{-1} )\), we can start by rewriting the terms with negative exponents:

1. **Rewrite the expression:**
\[
\frac{x^{-2} - y^{-2}}{x^{-1} + y^{-1}}
\]
can be rewritten as:
\[
\frac{\frac{1}{x^2} - \frac{1}{y^2}}{\frac{1}{x} + \frac{1}{y}}
\]

2. **Finding a common denominator:**
For the numerator \( \frac{1}{x^2} - \frac{1}{y^2} \), we can express it with a common denominator:
\[
\frac{y^2 - x^2}{x^2y^2} = \frac{(y-x)(y+x)}{x^2y^2}
\]

For the denominator \( \frac{1}{x} + \frac{1}{y} \):
\[
\frac{y + x}{xy}
\]

3. **Putting it together:**
Now substitute these back into the fraction:
\[
\frac{\frac{(y-x)(y+x)}{x^2y^2}}{\frac{x + y}{xy}} = \frac{(y-x)(y+x)}{x^2y^2} \cdot \frac{xy}{x+y}
\]

4. **Simplifying:**
The \(xy\) in the numerator will cancel with part of the denominator:
\[
\frac{(y-x)(y+x)}{xy(x+y)} = \frac{(y-x)(y+x)}{xy(x+y)}
\]

So, the simplified form of the expression \(( x^{-2} - y^{-2} ) \div ( x^{-1} + y^{-1} )\) is:

\[
\frac{(y-x)(y+x)}{xy(x+y)}
\]