Question
What's the error? A student said that $\frac{3^5}{9^5}$ is the same as $\frac{1}{15}$. What mistake has the student made?
✓
Answer
We have, $\frac{3^5}{9^5}=\frac{3^5}{\left(3^2\right)^5} \quad\left[\because 9=3 \times 3=3^2\right]$
$=\frac{3^5}{3^{10}}=\frac{1}{3^{10-5}}=\frac{1}{3^5} \quad\left[\because \frac{a^m}{b^n}=a^{m-n}\right]$
So, $\frac{1}{15}$ is not same as $\frac{1}{3^5}$ student has multiplied the base by its exponent. This is the error.
$=\frac{3^5}{3^{10}}=\frac{1}{3^{10-5}}=\frac{1}{3^5} \quad\left[\because \frac{a^m}{b^n}=a^{m-n}\right]$
So, $\frac{1}{15}$ is not same as $\frac{1}{3^5}$ student has multiplied the base by its exponent. This is the error.
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