Sure, let's solve the problem step by step.
We are asked to find [tex]\(\sqrt[8]{512}\)[/tex], which represents the eighth root of 512.
1. Understanding the Eighth Root:
The eighth root of a number [tex]\( n \)[/tex] is a value [tex]\( x \)[/tex] such that [tex]\( x^8 = n \)[/tex].
2. Setting Up the Equation:
[tex]\[
x^8 = 512
\][/tex]
3. Solving for [tex]\( x \)[/tex]:
We want to solve for [tex]\( x \)[/tex] such that raising [tex]\( x \)[/tex] to the power of 8 gives us 512.
4. Eighth Root Calculation:
The eighth root of 512 is approximately:
[tex]\[
\sqrt[8]{512} \approx 2.1810154653305154
\][/tex]
So, the eighth root of 512 is approximately [tex]\( 2.181 \)[/tex] (rounded to three decimal places).
