\( \sqrt[3]{7}=7^{\frac{1}{3}} \) A. \( \left(7^{\frac{1}{3}}\right)^{3}=7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}}=7 \cdot\left(\frac{1}{3}+\frac{1}{3}+\frac{1}{3}\right)=7 \cdot \frac{3}{3}=7 \cdot 1=7 \) B. \( \left(7^{\frac{1}{3}}\right)^{3}=7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}}=7 \cdot 7^{\frac{1}{3}}=3 \cdot \frac{1}{3} \cdot 7=1 \cdot 7=7 \) C. \( \left(7^{\frac{1}{3}}\right)^{3}=7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}}=7^{\frac{1}{3} \cdot \frac{1}{3} \cdot \frac{1}{3}}=7^{\frac{3}{3}}=7^{1}=7 \) D. \( \left(7^{\frac{1}{3}}\right)^{3}=7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}}=7^{\frac{1}{3}+\frac{1}{3}+\frac{1}{3}}=7^{\frac{3}{3}}=7^{1}=7 \)

Let's analyze each option step by step to determine if they correctly demonstrate that \( \left(7^{\frac{1}{3}}\right)^{3} = 7 \). ### A. \[ \left(7^{\frac{1}{3}}\right)^{3} = 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} = 7 \cdot \left(\frac{1}{3}+\frac{1}{3}+\frac{1}{3}\right) = 7 \cdot \frac{3}{3} = 7 \cdot 1 = 7 \] - This option correctly shows that \( \left(7^{\frac{1}{3}}\right)^{3} = 7 \) using the property of exponents. ### B. \[ \left(7^{\frac{1}{3}}\right)^{3} = 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} = 7 \cdot 7^{\frac{1}{3}} = 3 \cdot \frac{1}{3} \cdot 7 = 1 \cdot 7 = 7 \] - This option contains an error in the step \( 7 \cdot 7^{\frac{1}{3}} \). It should not equal \( 3 \cdot \frac{1}{3} \cdot 7 \). Therefore, this option is incorrect. ### C. \[ \left(7^{\frac{1}{3}}\right)^{3} = 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} = 7^{\frac{1}{3} \cdot \frac{1}{3} \cdot \frac{1}{3}} = 7^{\frac{3}{3}} = 7^{1} = 7 \] - This option incorrectly states \( 7^{\frac{1}{3} \cdot \frac{1}{3} \cdot \frac{1}{3}} \) as it should be \( 7^{\frac{1}{3} + \frac{1}{3} + \frac{1}{3}} \). Thus, this option is incorrect. ### D. \[ \left(7^{\frac{1}{3}}\right)^{3} = 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} = 7^{\frac{1}{3}+\frac{1}{3}+\frac{1}{3}} = 7^{\frac{3}{3}} = 7^{1} = 7 \] - This option correctly uses the property of exponents to show that \( \left(7^{\frac{1}{3}}\right)^{3} = 7 \). ### Conclusion: The correct options that demonstrate \( \left(7^{\frac{1}{3}}\right)^{3} = 7 \) are **A** and **D**. Options **B** and **C** contain errors in their calculations.