To express [tex]\(\sqrt[5]{13^9}\)[/tex] in the form [tex]\(a^{\frac{b}{c}}\)[/tex], follow these steps:
1. Identify the base of the exponential expression inside the radical. Here, the base is [tex]\(13\)[/tex].
2. Identify the exponent of the base within the radical. For [tex]\(13^9\)[/tex], the exponent is [tex]\(9\)[/tex].
3. Recognize the radical's index. The fifth root is represented by [tex]\(5\)[/tex], so the radical's index is [tex]\(5\)[/tex].
To convert the radical expression into an exponent form, the general rule is to use the index of the root as the denominator and the exponent of the base as the numerator:
[tex]\[
\sqrt[n]{x^m} = x^{\frac{m}{n}}
\][/tex]
Applying this rule, we get:
[tex]\[
\sqrt[5]{13^9} = 13^{\frac{9}{5}}
\][/tex]
Thus, in the expression [tex]\(a^{\frac{b}{c}}\)[/tex]:
- [tex]\(a\)[/tex] (the base) is [tex]\(13\)[/tex],
- [tex]\(b\)[/tex] (the numerator of the exponent) is [tex]\(9\)[/tex],
- [tex]\(c\)[/tex] (the denominator of the exponent) is [tex]\(5\)[/tex].
Therefore:
- [tex]\(a = 13\)[/tex],
- [tex]\(b = 9\)[/tex],
- [tex]\(c = 5\)[/tex].
So, the completed sentence will be:
The expression above can also be written in the form [tex]\(a^{\frac{b}{c}}\)[/tex]. For this expression, [tex]\(a = 13\)[/tex], [tex]\(b = 9\)[/tex], and [tex]\(c = 5\)[/tex].
