Question
a) \( \frac{3^{\frac{2}{5}} \cdot 3^{-\frac{4}{5}}}{3^{\frac{8}{5}}} \)
Ask by Hall Turner. in Canada
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The simplified expression is \( \frac{1}{9} \).
Solution
Step 1: Combine the numerator using exponent rules.
\[
3^{\frac{2}{5}} \cdot 3^{-\frac{4}{5}} = 3^{\frac{2}{5} + \left(-\frac{4}{5}\right)} = 3^{-\frac{2}{5}}
\]
Step 2: Divide the resulting expression by \( 3^{\frac{8}{5}} \).
\[
\frac{3^{-\frac{2}{5}}}{3^{\frac{8}{5}}} = 3^{-\frac{2}{5} - \frac{8}{5}} = 3^{-\frac{10}{5}}
\]
Step 3: Simplify the exponent.
\[
3^{-\frac{10}{5}} = 3^{-2}
\]
Step 4: Express the final answer as a fraction.
\[
3^{-2} = \frac{1}{3^2} = \frac{1}{9}
\]
The simplified expression is \( \frac{1}{9} \).
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Simplify this solution Extra Insights
First, let's simplify the expression using the properties of exponents. We can combine the exponents in the numerator: \[ 3^{\frac{2}{5}} \cdot 3^{-\frac{4}{5}} = 3^{\frac{2}{5} - \frac{4}{5}} = 3^{-\frac{2}{5}} \] Now, substituting this back into the expression gives us: \[ \frac{3^{-\frac{2}{5}}}{3^{\frac{8}{5}}} \] Next, we apply the quotient rule, which states that when dividing powers with the same base, we subtract the exponents: \[ 3^{-\frac{2}{5} - \frac{8}{5}} = 3^{-\frac{10}{5}} = 3^{-2} \] Finally, we can express \(3^{-2}\) as a fraction: \[ 3^{-2} = \frac{1}{3^2} = \frac{1}{9} \] So, the final answer is: \[ \frac{1}{9} \]
