Pregunta
Use radical notation to write the expression. Simplify if possible. Assume that all variables represent positive real numbers. \[ \begin{array}{l}(5 x+3)^{\frac{3}{4}} \\ \text { Select the correct choice below and, if necessary, fill in the answer box to complete your choice. } \\ \text { A. }(5 x+3)^{\frac{3}{4}}=\square \\ (\text { Simplify your answer. Type an exact answer, using radicals as needed.) } \\ \text { B. The answer is not a real number. }\end{array} \]
Ask by Pritchard Joseph. in the United States
Mar 15,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
\[
(5x + 3)^{\frac{3}{4}} = \sqrt[4]{(5x + 3)^3}
\]
Solución
We start with the expression
\[
(5x+3)^{\frac{3}{4}}
\]
Recall that an exponent of the form \(\frac{m}{n}\) can be rewritten using radicals as
\[
a^{\frac{m}{n}}=\sqrt[n]{a^m}
\]
Here, \(m=3\) and \(n=4\). Thus, we can rewrite the expression as
\[
(5x+3)^{\frac{3}{4}}=\sqrt[4]{(5x+3)^3}
\]
Since \(5x+3\) is positive (by assumption), the radical expression is valid and cannot be simplified any further.
Therefore, the answer is
\[
\sqrt[4]{(5x+3)^3}
\]
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Simplifique esta solución Extra Insights
To express \( (5x + 3)^{\frac{3}{4}} \) using radical notation, we can rewrite it as: \[ (5x + 3)^{\frac{3}{4}} = \sqrt[4]{(5x + 3)^3} \] This expresses the problem in radical form, where we have the fourth root of \((5x + 3)^3\). To simplify, we won't simplify this expression further since \( (5x + 3)^3 \) does not have any common factors that would allow for further reduction when taking the fourth root. Thus the answer in radical notation is: \[ \sqrt[4]{(5x + 3)^3} \]
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