To simplify the expression [tex]\(\left(4^{\frac{1}{5}}\right)^5\)[/tex], we will follow the properties of exponents step by step.
1. Understand the Property of Exponents:
The given expression involves raising a power to another power: [tex]\(\left(a^m\right)^n = a^{mn}\)[/tex].
2. Apply the Property:
Here, the base [tex]\(a\)[/tex] is 4, [tex]\(m\)[/tex] is [tex]\(\frac{1}{5}\)[/tex], and [tex]\(n\)[/tex] is 5. Using the property, we have:
[tex]\[
\left(4^{\frac{1}{5}}\right)^5 = 4^{\frac{1}{5} \cdot 5}
\][/tex]
3. Simplify the Exponent:
Next, simplify the exponent multiplication:
[tex]\[
\frac{1}{5} \cdot 5 = 1
\][/tex]
Therefore:
[tex]\[
4^{\frac{1}{5} \cdot 5} = 4^1
\][/tex]
4. Compute the Final Result:
Finally, compute [tex]\(4^1\)[/tex]:
[tex]\[
4^1 = 4
\][/tex]
Now, this is the theoretical way to simplify the expression. However, when calculated precisely, the intermediate value [tex]\(4^{\frac{1}{5}}\)[/tex] is approximately 1.3195079107728942, and raising it back to the 5th power yields a result very close, but not exactly equal, to 4. The numerical output of this operation is approximately 3.999999999999999 due to computational rounding.
Thus, the simplified form of [tex]\(\left(4^{\frac{1}{5}}\right)^5\)[/tex] is 4, but when computed numerically, it slightly differs due to the precision of calculations.
