Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Our platform connects you with professionals ready to provide precise answers to all your questions in various areas of expertise. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To transform the expression \(\left(\sqrt[5]{x^7}\right)^3\) into an expression with a rational exponent, we should follow a step-by-step approach that involves understanding and applying the properties of exponents and radicals.
Firstly, recognize that the fifth root of \(x^7\) can be expressed as \(x^{7/5}\). This stems from the general property that the \(n\)-th root of \(x^m\) can be written as \(x^{m/n}\). Thus,
[tex]\[ \sqrt[5]{x^7} = x^{7/5}. \][/tex]
Next, we need to raise this result to the power of 3. According to the rules of exponents, specifically the power of a power rule, \(\left( x^a \right)^b = x^{a \cdot b}\). Therefore,
[tex]\[ \left( x^{7/5} \right)^3 = x^{(7/5) \cdot 3}. \][/tex]
Now, compute the product of the exponents:
[tex]\[ (7/5) \cdot 3 = 21/5. \][/tex]
Putting it all together, the expression \(\left(\sqrt[5]{x^7}\right)^3\) simplifies to \(x^{21/5}\).
So, the transformation of [tex]\(\left(\sqrt[5]{x^7}\right)^3\)[/tex] results in [tex]\(x^{21/5}\)[/tex] expressed with a rational exponent.
Firstly, recognize that the fifth root of \(x^7\) can be expressed as \(x^{7/5}\). This stems from the general property that the \(n\)-th root of \(x^m\) can be written as \(x^{m/n}\). Thus,
[tex]\[ \sqrt[5]{x^7} = x^{7/5}. \][/tex]
Next, we need to raise this result to the power of 3. According to the rules of exponents, specifically the power of a power rule, \(\left( x^a \right)^b = x^{a \cdot b}\). Therefore,
[tex]\[ \left( x^{7/5} \right)^3 = x^{(7/5) \cdot 3}. \][/tex]
Now, compute the product of the exponents:
[tex]\[ (7/5) \cdot 3 = 21/5. \][/tex]
Putting it all together, the expression \(\left(\sqrt[5]{x^7}\right)^3\) simplifies to \(x^{21/5}\).
So, the transformation of [tex]\(\left(\sqrt[5]{x^7}\right)^3\)[/tex] results in [tex]\(x^{21/5}\)[/tex] expressed with a rational exponent.
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.