- Use the property \(\sqrt[n]{x^m} = x^{m/n}\) to convert each radical form into a power of 5.
- Determine the exponent for each term: \(5^1\) for \(5\), \(5^{1/2}\) for \(\sqrt{5}\), \(5^{1/3}\) for \(\sqrt[3]{5}\), \(5^{2/3}\) for \(\sqrt[3]{25}\), and \(5^{3/2}\) for \(\sqrt{125}\).
- The final mappings are: \(5=5^1\), \(\sqrt{5}=5^{1/2}\), \(\sqrt[3]{5}=5^{1/3}\), \(\sqrt[3]{25}=5^{2/3}\), and \(\sqrt{125}=5^{3/2}\).
- Final answer: \(\boxed{5=5^1, \sqrt{5}=5^{1/2}, \sqrt[3]{5}=5^{1/3}, \sqrt[3]{25}=5^{2/3}, \sqrt{125}=5^{3/2}}\)
Explanation
- Analyzing the Expressions
We need to represent each of the given expressions as a power of 5 with a rational exponent. The expressions provided are \(5\), \(\sqrt{5}\), \(\sqrt[3]{5}\), \(\sqrt[3]{25}\), and \(\sqrt{125}\). We recall the definition of rational exponents: \(\sqrt[n]{x^m} = x^{\frac{m}{n}}\), where \(x\) is the base, \(m\) is the power, and \(n\) is the index of the root. We also know that \(5 = 5^1\). Our objective is to determine the exponent \(k\) for each expression such that it equals \(5^k\).
- Converting Radicals to Exponential Forms
We convert each term into the form \(5^k\):
- For \(5\): Since \(5 = 5^1\), the exponent is \(1\).
- For \(\sqrt{5}\): The square root is equivalent to a root with index \(2\), so \(\sqrt{5} = 5^{\frac{1}{2}}\).
- For \(\sqrt[3]{5}\): The cube root is equivalent to a root with index \(3\), so \(\sqrt[3]{5} = 5^{\frac{1}{3}}\).
- For \(\sqrt[3]{25}\): Since \(25 = 5^2\), we have \(\sqrt[3]{25} = \sqrt[3]{5^2} = 5^{\frac{2}{3}}\).
- For \(\sqrt{125}\): Since \(125 = 5^3\), we have \(\sqrt{125} = \sqrt{5^3} = 5^{\frac{3}{2}}\).
- Conclusion and Final Matches
Based on the conversions above, we match each radical or simplified form to its corresponding rational exponent:
- \(5 = 5^1\)
- \(\sqrt{5} = 5^{\frac{1}{2}}\)
- \(\sqrt[3]{5} = 5^{\frac{1}{3}}\)
- \(\sqrt[3]{25} = 5^{\frac{2}{3}}\)
- \(\sqrt{125} = 5^{\frac{3}{2}}\)
These equalities confirm the relationships between the radical notations and the rational exponent notation. Thus, the matches are:
\(5 \to 5^1\)
\(\sqrt{5} \to 5^{\frac{1}{2}}\)
\(\sqrt[3]{5} \to 5^{\frac{1}{3}}\)
\(\sqrt[3]{25} \to 5^{\frac{2}{3}}\)
\(\sqrt{125} \to 5^{\frac{3}{2}}\)
The solution is complete.
Examples
In fields like computer science and digital signal processing, working with roots and exponents is common, for instance, when scaling dimensions in image processing or calculating signal decay rates. Understanding how to convert radicals like \(\sqrt{2}\) or \(\sqrt[3]{4}\) into powers like \(2^{0.5}\) and \(2^{2/3}\) allows for easier computation and manipulation of these values using standard algebraic rules, simplifying complex engineering problems significantly.