QuestionJune 17, 2025

Write the following in simplified radical form. sqrt [5](64x^7) Assume that the variable represents a positive real number. square

Write the following in simplified radical form. sqrt [5](64x^7) Assume that the variable represents a positive real number. square
Write the following in simplified radical form. sqrt [5](64x^7) Assume that the variable represents a positive real number. square

Solution
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Answer

2^{6/5} \cdot x \cdot \sqrt[5]{x^2} Explanation 1. Simplify the expression inside the radical Recognize that 64 = 2^6 and rewrite x^7 as (x^5) \cdot x^2. Thus, \sqrt[5]{64x^7} = \sqrt[5]{2^6 \cdot x^5 \cdot x^2}. 2. Apply the property of radicals Use the property \sqrt[n]{a^m} = a^{m/n} to simplify: \sqrt[5]{2^6} = 2^{6/5} and \sqrt[5]{x^5} = x^{5/5} = x. 3. Combine simplified terms Combine the results: 2^{6/5} \cdot x \cdot \sqrt[5]{x^2}.

Explanation

1. Simplify the expression inside the radical<br /> Recognize that $64 = 2^6$ and rewrite $x^7$ as $(x^5) \cdot x^2$. Thus, $\sqrt[5]{64x^7} = \sqrt[5]{2^6 \cdot x^5 \cdot x^2}$.<br /><br />2. Apply the property of radicals<br /> Use the property $\sqrt[n]{a^m} = a^{m/n}$ to simplify: $\sqrt[5]{2^6} = 2^{6/5}$ and $\sqrt[5]{x^5} = x^{5/5} = x$.<br /><br />3. Combine simplified terms<br /> Combine the results: $2^{6/5} \cdot x \cdot \sqrt[5]{x^2}$.
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