To rewrite the radical term [tex]\(\sqrt[8]{a^7}\)[/tex] in rational exponent form, follow these steps:
1. Understand the radical notation: The expression [tex]\(\sqrt[8]{a^7}\)[/tex] represents the 8th root of [tex]\(a^7\)[/tex].
2. Convert the radical into exponent notation: In general, the [tex]\(n\)[/tex]-th root of a term can be written as an exponent with a fractional power. Specifically, [tex]\(\sqrt[n]{x} = x^{1/n}\)[/tex].
3. Apply the rule to the given expression: For the expression [tex]\(\sqrt[8]{a^7}\)[/tex], the index of the root is 8, and the radicand is [tex]\(a^7\)[/tex]. So we can rewrite the expression using fractional exponents.
4. Combine the exponents: The expression [tex]\(\sqrt[8]{a^7}\)[/tex] will be written as [tex]\(a\)[/tex] raised to the power of [tex]\(\frac{7}{8}\)[/tex]. This is because:
[tex]\[
\sqrt[8]{a^7} = a^{7/8}
\][/tex]
5. Conclusion: Therefore, the radical expression [tex]\(\sqrt[8]{a^7}\)[/tex] in rational exponent form is:
[tex]\[
a^{0.875}
\][/tex]
Thus, [tex]\(\sqrt[8]{a^7} = a^{0.875}\)[/tex].
