- The definition of the cube root \(\sqrt[3]{x} = y \iff y^3 = x\) confirms the equivalence when applied to \(\left(7^{\frac{4}{3}}\right)^3 = 7^4\).
- The property of rational exponents \(\sqrt[n]{a^m} = (a^m)^{\frac{1}{n}}\) justifies the equivalence shown by \(7^{\frac{4}{3}} = (7^4)^{\frac{1}{3}}\).
- Other options involve valid exponent arithmetic but do not serve to define or demonstrate the relationship between the radical and the exponent.
- The correct options are: \(\boxed{7^{\frac{4}{3}} = (7^4)^{\frac{1}{3}}}\) and \(\boxed{\left(7^{\frac{4}{3}}\right)^3 = 7^{\frac{4}{3} \cdot 3} = 7^4}\).
Explanation
- Analyzing the Definitions of Radicals and Exponents
The goal is to determine which of the provided expressions correctly demonstrate the equivalence \(\sqrt[3]{7^4} = 7^{\frac{4}{3}}\).
First, consider the definition of the n-th root. The expression \(\sqrt[n]{x} = y\) is equivalent to \(y^n = x\). Applying this to our case where \(n=3\) and \(x=7^4\), we have \(\sqrt[3]{7^4} = y \iff y^3 = 7^4\). If we let \(y = 7^{\frac{4}{3}}\), then the condition becomes \((7^{\frac{4}{3}})^3 = 7^4\). Using the power of a power rule, \((a^m)^n = a^{m \cdot n}\), we get \(7^{\frac{4}{3} \cdot 3} = 7^4\), which is a valid confirmation.
Second, consider the definition of rational exponents, which states that \(\sqrt[n]{a^m} = a^{\frac{m}{n}}\). By definition, \(\sqrt[n]{x} = x^{\frac{1}{n}}\). Thus, \(\sqrt[3]{7^4} = (7^4)^{\frac{1}{3}}\). Combined with the property \((a^m)^n = a^{m \cdot n}\), this shows \((7^4)^{\frac{1}{3}} = 7^{4 \cdot \frac{1}{3}} = 7^{\frac{4}{3}}\). Therefore, the equation \(7^{\frac{4}{3}} = (7^4)^{\frac{1}{3}}\) is a valid demonstration of the property of rational exponents applied to this expression.
- Evaluating the Remaining Options
We evaluate the remaining options:
- \(7^{\frac{4}{3}} \cdot 7^{\frac{8}{3}} = 7^{\frac{4}{3} + \frac{8}{3}} = 7^4\): While this is a true mathematical statement, it does not demonstrate the relationship between the radical \(\sqrt[3]{7^4}\) and the exponent \(7^{\frac{4}{3}}\). It is simply an example of exponent multiplication.
- \(\frac{7^{\frac{19}{3}}}{7^{\frac{7}{3}}} = 7^{\frac{19}{3} - \frac{7}{3}} = 7^4\): Similarly, this is a correct arithmetic calculation, but it does not link the radical \(\sqrt[3]{7^4}\) to the power \(7^{\frac{4}{3}}\).
- \(7^{\frac{2}{3}} \cdot 7^{\frac{10}{3}} = 7^{\frac{2}{3} + \frac{10}{3}} = 7^4\): Again, this is true but does not define or demonstrate the given radical-exponent relationship.
Therefore, the two properties that specifically explain the transition from radical form to rational exponent form are \(7^{\frac{4}{3}} = (7^4)^{\frac{1}{3}}\) and \(\left(7^{\frac{4}{3}}\right)^3 = 7^4\).
- Final Conclusion
Based on the analysis, the two correct properties are \(\left(7^{\frac{4}{3}}\right)^3 = 7^{\frac{4}{3} \cdot 3} = 7^4\) and \(7^{\frac{4}{3}} = (7^4)^{\frac{1}{3}}\). The final answer is the selection of these two specific expressions.
Examples
In real-world applications, such as calculating the growth of investments with compound interest or determining the dimensions of geometric objects, we often need to switch between radical and exponential notation to simplify complex equations. For example, if you are calculating the side length of a cube with volume \(V = 7^4\), you would find it as \(V^{\frac{1}{3}} = (7^4)^{\frac{1}{3}} = 7^{\frac{4}{3}}\). Understanding these conversion rules makes solving such problems much more efficient.