To write the expression [tex]\( z^{\frac{4}{7}} \)[/tex] as a radical expression, we need to understand how to interpret the exponent [tex]\(\frac{4}{7}\)[/tex].
1. Understand the Exponent: The exponent [tex]\(\frac{4}{7}\)[/tex] can be broken down into two parts:
- The numerator [tex]\(4\)[/tex] represents a power.
- The denominator [tex]\(7\)[/tex] represents a root.
2. Express as a Power and Root: An exponent of the form [tex]\(\frac{4}{7}\)[/tex] suggests that we take the 7th root (denominator) and raise the result to the 4th power (numerator). Mathematically, this can be written as:
[tex]\[
z^{\frac{4}{7}} = \left(z^4\right)^{\frac{1}{7}}
\][/tex]
3. Convert to a Radical Form: The expression [tex]\(\left(z^4\right)^{\frac{1}{7}}\)[/tex] can be rewritten using radical notation. The [tex]\(\frac{1}{7}\)[/tex] exponent indicates the 7th root:
[tex]\[
\left(z^4\right)^{\frac{1}{7}} = \sqrt[7]{z^4}
\][/tex]
Thus, the expression [tex]\( z^{\frac{4}{7}} \)[/tex] as a radical expression is:
[tex]\[
\sqrt[7]{z^4}
\][/tex]
This result shows that [tex]\( z^{\frac{4}{7}} \)[/tex] can be interpreted as the 7th root of [tex]\( z \)[/tex] raised to the 4th power.
