The problem is to complete a table that contains expressions written in radical notation, rational exponent notation, and their decimal approximations. We are given some entries and need to fill in the missing ones.

AlgebraExponentsRadicalsSimplificationRational Exponents
2025/7/3

1. Problem Description

The problem is to complete a table that contains expressions written in radical notation, rational exponent notation, and their decimal approximations. We are given some entries and need to fill in the missing ones.

2. Solution Steps

Row 2: Given 1754\sqrt[4]{17^5} in radical notation.
To convert to rational exponent form, we use the rule: amn=amn\sqrt[n]{a^m} = a^{\frac{m}{n}}.
Therefore, 1754=1754\sqrt[4]{17^5} = 17^{\frac{5}{4}}.
Using a calculator, 175481.6817^{\frac{5}{4}} \approx 81.68.
Row 3: Given 1545\sqrt[5]{15^4} in radical notation.
To convert to rational exponent form, we use the rule: amn=amn\sqrt[n]{a^m} = a^{\frac{m}{n}}.
Therefore, 1545=1545\sqrt[5]{15^4} = 15^{\frac{4}{5}}.
Using a calculator, 15457.2315^{\frac{4}{5}} \approx 7.23.
Row 4: Given 114311^{\frac{4}{3}} in rational exponent notation.
To convert to radical form, we use the rule: amn=amna^{\frac{m}{n}} = \sqrt[n]{a^m}.
Therefore, 1143=114311^{\frac{4}{3}} = \sqrt[3]{11^4}.
Using a calculator, 114323.7211^{\frac{4}{3}} \approx 23.72.
Row 5: Given 2852^{\frac{8}{5}} in rational exponent notation.
To convert to radical form, we use the rule: amn=amna^{\frac{m}{n}} = \sqrt[n]{a^m}.
Therefore, 285=2852^{\frac{8}{5}} = \sqrt[5]{2^8}.
Using a calculator, 2853.032^{\frac{8}{5}} \approx 3.03.

3. Final Answer

Row 2: Written using rational exponents: 175417^{\frac{5}{4}}, Evaluated to two decimal places: 81.68
Row 3: Written using rational exponents: 154515^{\frac{4}{5}}, Evaluated to two decimal places: 7.23
Row 4: Written in radical notation: 1143\sqrt[3]{11^4}, Evaluated to two decimal places: 23.72
Row 5: Written in radical notation: 285\sqrt[5]{2^8}, Evaluated to two decimal places: 3.03

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