Select the simplification that accurately explains the following statement. \( \sqrt[8]{7}=7^{\frac{1}{3}} \) A. \( \left(7^{\frac{1}{3}}\right)^{8}=7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}}=7 \cdot\left(\frac{1}{3}+\frac{1}{8}+\frac{1}{3}\right)=7 \cdot \frac{3}{3}=7 \cdot 1=7 \) 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. \( \left(7^{\frac{1}{3}}\right)^{8}=7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}}=7^{\frac{1}{3} \cdot \frac{1}{3}} \frac{1}{3}=7^{\frac{3}{3}}=7^{1}=7 \) D. \( \left(7^{\frac{1}{3}}\right)^{8}=7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}}=7^{\frac{1}{3}}+\frac{1}{3}+\frac{1}{3}=7^{\frac{3}{3}}=7^{1}=7 \)

To determine which simplification accurately explains the statement \( \sqrt[8]{7}=7^{\frac{1}{3}} \), we need to analyze the options provided. First, let's rewrite the left side of the equation: \[ \sqrt[8]{7} = 7^{\frac{1}{8}} \] Now, we need to compare this with the right side, which is \( 7^{\frac{1}{3}} \). Therefore, we are actually comparing \( 7^{\frac{1}{8}} \) with \( 7^{\frac{1}{3}} \). Next, we can analyze each option to see if they correctly represent the relationship between these two expressions. ### Option A: \[ \left(7^{\frac{1}{3}}\right)^{8} = 7^{\frac{8}{3}} \quad \text{(This is incorrect)} \] The simplification does not lead to \( 7 \). ### Option B: \[ \left(7^{\frac{1}{3}}\right)^{3} = 7^{1} = 7 \quad \text{(This is also incorrect)} \] This does not relate to \( 7^{\frac{1}{8}} \). ### Option C: \[ \left(7^{\frac{1}{3}}\right)^{8} = 7^{\frac{8}{3}} \quad \text{(This is incorrect)} \] Again, this does not lead to \( 7 \). ### Option D: \[ \left(7^{\frac{1}{3}}\right)^{8} = 7^{\frac{8}{3}} \quad \text{(This is incorrect)} \] This does not lead to \( 7 \). ### Conclusion: None of the options A, B, C, or D correctly explain the statement \( \sqrt[8]{7}=7^{\frac{1}{3}} \). In fact, \( 7^{\frac{1}{8}} \neq 7^{\frac{1}{3}} \). The correct relationship is that \( 7^{\frac{1}{8}} \) is less than \( 7^{\frac{1}{3}} \). Thus, the statement \( \sqrt[8]{7}=7^{\frac{1}{3}} \) is false, and none of the provided options accurately explain it.