To express the cube root of a number using a rational exponent, we need to understand the relationship between roots and exponents. Specifically, the cube root of a number [tex]\( x \)[/tex] can be written as [tex]\( x \)[/tex] raised to the power of [tex]\( \frac{1}{3} \)[/tex].
Here, we are given the cube root of 5, which can be written in radical form as [tex]\( \sqrt[3]{5} \)[/tex]. To rewrite this expression using a rational exponent, we replace the radical with an exponent. The cube root of 5 would be written as:
[tex]\[ \sqrt[3]{5} = 5^{\frac{1}{3}} \][/tex]
Therefore, the correct way to rewrite [tex]\( \sqrt[3]{5} \)[/tex] using a rational exponent is [tex]\( 5^{\frac{1}{3}} \)[/tex].
Given the choices:
- [tex]\( 5^{\frac{2}{3}} \)[/tex]
- [tex]\( 5^1 \)[/tex]
- [tex]\( 5^3 \)[/tex]
- [tex]\( 3^5 \)[/tex]
None of these choices match [tex]\( 5^{\frac{1}{3}} \)[/tex]. Thus, there seems to be an error in the answer choices provided. However, the correct way to express [tex]\( \sqrt[3]{5} \)[/tex] using a rational exponent is [tex]\( 5^{\frac{1}{3}} \)[/tex], even though it is not listed among the provided options.
