Answer:

To simplify the expression 11^(7/3) + 11^(-1/3), we can first rewrite the terms with the same base, which in this case is 11.

Using the property of exponents that states a^(m/n) = (a^m)^(1/n), we can rewrite the expression as:

11^(7/3) + 11^(-1/3) = (11^7)^(1/3) + (1/11)^(1/3)

Now, we can simplify each term separately:

(11^7)^(1/3) = 11^(7/3) = 11^2 _ 11^(1/3) = 121 _ (11^(1/3))

(1/11)^(1/3) = 1^(1/3) / 11^(1/3) = 1 / 11^(1/3)

Therefore, the simplified expression becomes:

121 * (11^(1/3)) + (1 / 11^(1/3))

Now, we have the terms with the same base, 11^(1/3). To combine them, we can multiply the fractions:

121 _ (11^(1/3)) + (1 / 11^(1/3)) = (121 _ 11^(1/3) + 1) / 11^(1/3)

This is the simplified form of the expression 11^(7/3) + 11^(-1/3).