Select the correct answer.Simplify the following expression.
A. 12
B. 81
C.
D.
The Correct Answer and Explanation is:
The given expression is:
[
\frac{3^{\frac{11}{5}}}{3^{-\frac{2}{5}}}
]
Step 1: Apply the quotient rule for exponents
The quotient rule for exponents states that:
[
\frac{a^m}{a^n} = a^{m-n}
]
In this case, the base is 3, so we apply this rule to subtract the exponents:
[
\frac{3^{\frac{11}{5}}}{3^{-\frac{2}{5}}} = 3^{\frac{11}{5} – \left(-\frac{2}{5}\right)} = 3^{\frac{11}{5} + \frac{2}{5}}
]
Step 2: Simplify the exponent
Now, simplify the exponent:
[
\frac{11}{5} + \frac{2}{5} = \frac{11 + 2}{5} = \frac{13}{5}
]
Thus, the expression becomes:
[
3^{\frac{13}{5}}
]
Step 3: Interpret the result
The expression ( 3^{\frac{13}{5}} ) means we are raising 3 to the power of ( \frac{13}{5} ). This is equivalent to:
[
3^{\frac{13}{5}} = \left( 3^{13} \right)^{\frac{1}{5}}
]
This means we first calculate ( 3^{13} ), then take the fifth root of that result.
Step 4: Calculate ( 3^{13} )
First, calculate ( 3^{13} ):
[
3^{13} = 1594323
]
Now, take the fifth root of 1594323. The fifth root of 1594323 is approximately 12.
Thus:
[
3^{\frac{13}{5}} \approx 12
]
Step 5: Conclusion
Therefore, the correct answer is:
[
\boxed{12}
]
This result comes from applying the exponent rules and simplifying the expression step by step.