Sure, let's convert the given expression [tex]\(\sqrt[9]{a^4}\)[/tex] into its exponential form step by step.
1. Understand the radical notation:
[tex]\(\sqrt[9]{a^4}\)[/tex] represents the ninth root of [tex]\(a^4\)[/tex]. In general, the nth root of a number [tex]\(x\)[/tex] can be expressed in exponential form as [tex]\(x^{1/n}\)[/tex].
2. Apply this to our expression:
The ninth root can be written as raising to the power of [tex]\(1/9\)[/tex]. Thus, [tex]\(\sqrt[9]{a^4}\)[/tex] is equivalent to [tex]\((a^4)^{1/9}\)[/tex].
3. Use the properties of exponents:
When you have a power raised to another power [tex]\((x^m)^n\)[/tex], you multiply the exponents [tex]\(m\)[/tex] and [tex]\(n\)[/tex]. Therefore, [tex]\((a^4)^{1/9}\)[/tex] can be simplified by multiplying the exponents:
[tex]\[
(a^4)^{1/9} = a^{4 \cdot \frac{1}{9}} = a^{\frac{4}{9}}
\][/tex]
Thus, the exponential form of [tex]\(\sqrt[9]{a^4}\)[/tex] is [tex]\(a^{\frac{4}{9}}\)[/tex].
