Answer :
Sure! Let's simplify the expression [tex]\(\sqrt[5]{x} \cdot \sqrt[5]{x} \cdot \sqrt[5]{x} \cdot \sqrt[5]{x}\)[/tex].
1. First, recognize that [tex]\(\sqrt[5]{x}\)[/tex] can be written in exponential form as [tex]\(x^{\frac{1}{5}}\)[/tex].
2. Thus, the expression becomes:
[tex]\[ x^{\frac{1}{5}} \cdot x^{\frac{1}{5}} \cdot x^{\frac{1}{5}} \cdot x^{\frac{1}{5}} \][/tex]
3. Using the property of exponents that states [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex], we can combine the exponents:
[tex]\[ x^{\frac{1}{5}} \cdot x^{\frac{1}{5}} \cdot x^{\frac{1}{5}} \cdot x^{\frac{1}{5}} = x^{\left(\frac{1}{5} + \frac{1}{5} + \frac{1}{5} + \frac{1}{5}\right)} \][/tex]
4. When we add the exponents, we get:
[tex]\[ \frac{1}{5} + \frac{1}{5} + \frac{1}{5} + \frac{1}{5} = \frac{4}{5} \][/tex]
5. Therefore, the simplified form of the expression is:
[tex]\[ \boxed{x^{\frac{4}{5}}} \][/tex]
Among the given choices, the correct one is:
[tex]\(x^{\frac{4}{5}}\)[/tex].
1. First, recognize that [tex]\(\sqrt[5]{x}\)[/tex] can be written in exponential form as [tex]\(x^{\frac{1}{5}}\)[/tex].
2. Thus, the expression becomes:
[tex]\[ x^{\frac{1}{5}} \cdot x^{\frac{1}{5}} \cdot x^{\frac{1}{5}} \cdot x^{\frac{1}{5}} \][/tex]
3. Using the property of exponents that states [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex], we can combine the exponents:
[tex]\[ x^{\frac{1}{5}} \cdot x^{\frac{1}{5}} \cdot x^{\frac{1}{5}} \cdot x^{\frac{1}{5}} = x^{\left(\frac{1}{5} + \frac{1}{5} + \frac{1}{5} + \frac{1}{5}\right)} \][/tex]
4. When we add the exponents, we get:
[tex]\[ \frac{1}{5} + \frac{1}{5} + \frac{1}{5} + \frac{1}{5} = \frac{4}{5} \][/tex]
5. Therefore, the simplified form of the expression is:
[tex]\[ \boxed{x^{\frac{4}{5}}} \][/tex]
Among the given choices, the correct one is:
[tex]\(x^{\frac{4}{5}}\)[/tex].
