To express [tex]\(\sqrt[4]{23}\)[/tex] as an exponent, we need to understand the concept of roots and how they relate to exponents.
1. The notation [tex]\(\sqrt[4]{23}\)[/tex] represents the 4th root of 23.
2. In general, the [tex]\(n\)[/tex]th root of a number [tex]\(a\)[/tex] can be expressed as [tex]\(a^{1/n}\)[/tex].
For our specific case:
- The 4th root of 23 means we have [tex]\(23^{1/4}\)[/tex].
Here is the step-by-step solution:
1. Identify the type of root:
- We are dealing with a 4th root, so [tex]\( n = 4 \)[/tex].
2. Write the expression in terms of exponents:
- The expression [tex]\(\sqrt[4]{23}\)[/tex] can be rewritten as [tex]\(23^{1/4}\)[/tex].
3. Determine the rational exponent:
- The exponent in this case is [tex]\( \frac{1}{4} \)[/tex].
Thus, the rational number that can be used as an exponent to rewrite [tex]\(\sqrt[4]{23}\)[/tex] is [tex]\(\frac{1}{4}\)[/tex].
When we convert [tex]\(\frac{1}{4}\)[/tex] to a decimal, we get 0.25. So, [tex]\( 23^{0.25} \)[/tex] is another equivalent way to write [tex]\(\sqrt[4]{23}\)[/tex].
Therefore, the rational number that could be used as an exponent to rewrite [tex]\(\sqrt[4]{23}\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] or 0.25.
