To simplify the given expression [tex]\(\sqrt[7]{x} \cdot \sqrt[7]{x}\)[/tex], we can use properties of exponents and radicals. Here is the step-by-step solution:
1. Convert the radicals to exponents: Recall that [tex]\(\sqrt[n]{x}\)[/tex] can be written as [tex]\(x^{1/n}\)[/tex]. So, [tex]\(\sqrt[7]{x}\)[/tex] is equivalent to [tex]\(x^{1/7}\)[/tex].
2. Substitute the radicals with exponents in the expression: The given expression now becomes:
[tex]\[
x^{1/7} \cdot x^{1/7}
\][/tex]
3. Apply the product of powers property: When multiplying two expressions with the same base, you add the exponents. The product of powers property is:
[tex]\[
x^a \cdot x^b = x^{a + b}
\][/tex]
In our case, [tex]\(a = \frac{1}{7}\)[/tex] and [tex]\(b = \frac{1}{7}\)[/tex]. Thus,
[tex]\[
x^{1/7} \cdot x^{1/7} = x^{(1/7) + (1/7)}
\][/tex]
4. Add the exponents: Now add the exponents:
[tex]\[
\frac{1}{7} + \frac{1}{7} = \frac{2}{7}
\][/tex]
5. Simplified form: Therefore, the simplified form of the expression is:
[tex]\[
x^{\frac{2}{7}}
\][/tex]
Let's match this result with the given options:
- [tex]\(x^{\frac{3}{7}}\)[/tex]
- [tex]\(x^{\frac{1}{7}}\)[/tex]
- [tex]\(x^{\frac{3}{21}}\)[/tex]
- [tex]\(\sqrt[21]{x}\)[/tex]
The correct answer is [tex]\(x^{\frac{2}{7}}\)[/tex]. However, it is not explicitly listed in the options. Given the correct simplification steps and the provided results, it is essential to point out that the options provided do not directly match the accurate answer. Nonetheless, our simplified form [tex]\(x^{\frac{2}{7}}\)[/tex] is indeed the correct simplification of the given expression.
