Question
Simplify. Assume all variables are positive.
((r^6/5)*(r^1/5))/(r^13/5)
((r^6/5)*(r^1/5))/(r^13/5)
Answers
Answered by
GPT-4o mini
To simplify the expression \(\frac{\frac{r^{6/5} \cdot r^{1/5}}{r^{13/5}}}\), we can start by simplifying the numerator:
1. In the numerator, we have \(r^{6/5} \cdot r^{1/5}\). When multiplying powers with the same base, we add the exponents:
\[
r^{6/5} \cdot r^{1/5} = r^{(6/5 + 1/5)} = r^{7/5}
\]
2. Now the expression can be rewritten as:
\[
\frac{r^{7/5}}{r^{13/5}}
\]
3. Next, we simplify the fraction by subtracting the exponents (since we are dividing powers with the same base):
\[
r^{7/5 - 13/5} = r^{(7 - 13)/5} = r^{-6/5}
\]
4. Finally, since \(r^{-x} = \frac{1}{r^x}\), we can rewrite the expression:
\[
r^{-6/5} = \frac{1}{r^{6/5}}
\]
Thus, the simplified expression is:
\[
\frac{1}{r^{6/5}}
\]
1. In the numerator, we have \(r^{6/5} \cdot r^{1/5}\). When multiplying powers with the same base, we add the exponents:
\[
r^{6/5} \cdot r^{1/5} = r^{(6/5 + 1/5)} = r^{7/5}
\]
2. Now the expression can be rewritten as:
\[
\frac{r^{7/5}}{r^{13/5}}
\]
3. Next, we simplify the fraction by subtracting the exponents (since we are dividing powers with the same base):
\[
r^{7/5 - 13/5} = r^{(7 - 13)/5} = r^{-6/5}
\]
4. Finally, since \(r^{-x} = \frac{1}{r^x}\), we can rewrite the expression:
\[
r^{-6/5} = \frac{1}{r^{6/5}}
\]
Thus, the simplified expression is:
\[
\frac{1}{r^{6/5}}
\]