Question
Write the rational expression as an equivalent expression with the indicated denominator.19/(z^2-7z)=?/z(z-7)(z-5)
Answers
GPT-4o mini
To rewrite the expression \(\frac{19}{z^2 - 7z}\) with the denominator \(z(z-7)(z-5)\), we first factor the original denominator:
\[
z^2 - 7z = z(z - 7)
\]
Now, we need to find out how to incorporate \((z - 5)\) into the new denominator \(z(z - 7)(z - 5)\).
To convert \(\frac{19}{z(z - 7)}\) into an equivalent expression with the desired denominator, we can multiply the numerator and the denominator by \((z - 5)\):
\[
\frac{19}{z(z - 7)} \cdot \frac{(z - 5)}{(z - 5)} = \frac{19(z - 5)}{z(z - 7)(z - 5)}
\]
Thus the equivalent expression is:
\[
\frac{19(z - 5)}{z(z - 7)(z - 5)}
\]
So, the final result is:
\[
\frac{19(z - 5)}{z(z - 7)(z - 5)}
\]
You can further simplify if necessary based on your context, but this is the equivalent expression needed.
\[
z^2 - 7z = z(z - 7)
\]
Now, we need to find out how to incorporate \((z - 5)\) into the new denominator \(z(z - 7)(z - 5)\).
To convert \(\frac{19}{z(z - 7)}\) into an equivalent expression with the desired denominator, we can multiply the numerator and the denominator by \((z - 5)\):
\[
\frac{19}{z(z - 7)} \cdot \frac{(z - 5)}{(z - 5)} = \frac{19(z - 5)}{z(z - 7)(z - 5)}
\]
Thus the equivalent expression is:
\[
\frac{19(z - 5)}{z(z - 7)(z - 5)}
\]
So, the final result is:
\[
\frac{19(z - 5)}{z(z - 7)(z - 5)}
\]
You can further simplify if necessary based on your context, but this is the equivalent expression needed.