To find the value of [tex]\(5^{\frac{2}{3}}\)[/tex], follow these steps:
1. Understanding the Fractional Exponent:
- The expression [tex]\(5^{\frac{2}{3}}\)[/tex] can be interpreted as raising 5 to a power that is a fraction.
- Fractional exponents represent roots and powers. In this case, [tex]\(\frac{2}{3}\)[/tex] can be broken down into a root and a power.
- Specifically, [tex]\(5^{\frac{2}{3}} = (5^2)^{\frac{1}{3}}\)[/tex] or equivalently [tex]\((5^{\frac{1}{3}})^2\)[/tex]. Both interpretations are valid and will yield the same result.
2. Breaking it down:
- First, consider the root part. The exponent [tex]\( \frac{1}{3} \)[/tex] implies taking the cube root.
- Then, consider the power part. The exponent 2 implies squaring the number.
3. Calculate the Cube Root:
- Find the cube root of 5, which is [tex]\( 5^{\frac{1}{3}} \)[/tex].
4. Square the Result:
- Once you have [tex]\( 5^{\frac{1}{3}} \)[/tex], square this result to get [tex]\( (5^{\frac{1}{3}})^2 \)[/tex].
5. Final Value:
- Combining these operations, after computing the cube root of 5 and then squaring that result, we get the final value.
After completing these steps and calculations, the result is approximately:
[tex]\[ 5^{\frac{2}{3}} \approx 2.924017738212866 \][/tex]
Thus, the value of [tex]\(5^{\frac{2}{3}}\)[/tex] is approximately 2.924017738212866.
