To determine the correct simplification of the expression [tex]\(\sqrt[3]{7} = 7^{\frac{1}{3}}\)[/tex], let's examine each option:
### Option A:
[tex]\[
\left(7^{\frac{1}{3}}\right)^3 = 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} = 7^{\frac{1}{3} \cdot \frac{1}{3} \cdot \frac{1}{3}} = 7^{\frac{3}{3}} = 7^1 = 7
\][/tex]
This option incorrectly simplifies the exponents. The correct multiplication for exponents should be addition, not multiplication.
### Option B:
[tex]\[
\left(7^{\frac{1}{3}}\right)^3 = 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} = 7 \cdot 7^{\frac{1}{3}} = 3 \cdot \frac{1}{3} \cdot 7 = 1 \cdot 7 = 7
\][/tex]
This option contains errors in both the simplification process and the application of exponent rules.
### Option C:
[tex]\[
\left(7^{\frac{1}{3}}\right)^3 = 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} = 7 \cdot\left(\frac{1}{3} + \frac{1}{3} + \frac{1}{3}\right) = 7 \cdot \frac{3}{3} = 7 \cdot 1 = 7
\][/tex]
This option makes a logical error by mixing up the base multiplication and the exponent addition in an incorrect manner.
### Option D:
[tex]\[
\left(7^{\frac{1}{3}}\right)^3 = 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} = 7^{\frac{1}{3} + \frac{1}{3} + \frac{1}{3}} = 7^{\frac{3}{3}} = 7^1 = 7
\][/tex]
This option correctly applies the rule for multiplying exponents: when multiplying like bases, you add the exponents. The exponent [tex]\(\frac{1}{3} + \frac{1}{3} + \frac{1}{3}\)[/tex] simplifies to 1.
Thus, after evaluating all options, the correct simplification is:
[tex]\[
\left(7^{\frac{1}{3}}\right)^3 = 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} = 7^{\frac{1}{3} + \frac{1}{3} + \frac{1}{3}} = 7^{\frac{3}{3}} = 7^1 = 7
\][/tex]
The correct answer is option D.
