To evaluate [tex]\(64^{1/3}\)[/tex], let's follow the steps to find the cube root of 64.
1. Identify the base and the exponent: Here, the base is 64, and the exponent is [tex]\(\frac{1}{3}\)[/tex].
2. Understanding the meaning of the exponent: The exponent [tex]\(\frac{1}{3}\)[/tex] signifies the cube root of 64. Mathematically, [tex]\(64^{1/3}\)[/tex] is equivalent to [tex]\(\sqrt[3]{64}\)[/tex].
3. Determine the value of [tex]\(\sqrt[3]{64}\)[/tex]: The goal is to find a number which, when multiplied by itself three times, equals 64.
To verify:
[tex]\[
4 \times 4 \times 4 = 4^3 = 64
\][/tex]
Therefore, [tex]\( \sqrt[3]{64} = 4\)[/tex].
4. Derive the final result: After confirming the value, we conclude that:
[tex]\[
64^{1/3} \approx 4
\][/tex]
However, note that there can be slight numerical inaccuracies in calculations provided by most software tools. For this specific calculation, the result is found to be approximately:
[tex]\[
64^{1/3} \approx 3.9999999999999996
\][/tex]
This final approximation closely aligns with the exact value of 4.
