Answered

Simplify the following expression:

[tex]\[ 3^{\frac{11}{5}} \div 3^{-\frac{2}{5}} \][/tex]

A. [tex]\(\frac{1}{81}\)[/tex]
B. [tex]\(\frac{1}{12}\)[/tex]
C. 81
D. 12



Answer :

Certainly! Let's simplify the given expression step-by-step:

We start with the expression:
[tex]\[ 3^{\frac{11}{5}} \div 3^{-\frac{2}{5}} \][/tex]

Recall the property of exponents that states:

[tex]\[ a^m \div a^n = a^{m-n} \][/tex]

Using this property, we can combine the two powers of 3 by subtracting the exponents:

[tex]\[ 3^{\frac{11}{5}} \div 3^{-\frac{2}{5}} = 3^{\left(\frac{11}{5} - \left(-\frac{2}{5}\right)\right)} \][/tex]

Next, simplify the exponent:

[tex]\[ \frac{11}{5} - \left(-\frac{2}{5}\right) = \frac{11}{5} + \frac{2}{5} \][/tex]

Now, add the fractions:

[tex]\[ \frac{11}{5} + \frac{2}{5} = \frac{11 + 2}{5} = \frac{13}{5} \][/tex]

So, the expression simplifies to:

[tex]\[ 3^{\frac{13}{5}} \][/tex]

To further simplify, we can convert [tex]\(\frac{13}{5}\)[/tex] to a decimal. This gives us:

[tex]\[ 3^{2.6} \][/tex]

We need to find the numerical value of [tex]\(3^{2.6}\)[/tex]. Calculating this value, we get approximately:

[tex]\[ 3^{2.6} \approx 17.398638404385867 \][/tex]

This value does not directly match any of the provided options (A. [tex]\(\frac{1}{81}\)[/tex], B. [tex]\(\frac{1}{12}\)[/tex], C. 81, D. 12)).

Upon re-examining the given options, none of them approximate [tex]\(3^{2.6}\)[/tex] closely. Therefore, the correct simplification of the expression is:

[tex]\[ 3^{2.6} \approx 17.398638404385867 \][/tex]

None of the provided multiple-choice answers (A, B, C, D) are correct based on our simplification and calculation.