Let's simplify the expression step-by-step:
[tex]\[
\left(x^{\frac{2}{3}}\right)^{\frac{8}{8}}
\][/tex]
1. Understand the given expression:
The expression inside the parentheses, [tex]\( x^{\frac{2}{3}} \)[/tex], indicates that [tex]\( x \)[/tex] is raised to the power of [tex]\( \frac{2}{3} \)[/tex].
The exponent outside the parentheses is [tex]\( \frac{8}{8} \)[/tex].
2. Simplify the outer exponent:
The exponent [tex]\( \frac{8}{8} \)[/tex] simplifies to 1, because anything divided by itself is 1.
So, the expression now is:
[tex]\[
\left(x^{\frac{2}{3}}\right)^1
\][/tex]
3. Apply the power rule of exponents:
When an exponent is raised to another exponent, you multiply the exponents. Here, we simply have [tex]\( x^{\frac{2}{3}} \)[/tex] raised to the power of 1.
Simplifying this, we get:
[tex]\[
x^{\frac{2}{3}}
\][/tex]
4. Final Simplification:
The simplified form of [tex]\( x^{\frac{2}{3}} \)[/tex] remains [tex]\( x^{\frac{2}{3}} \)[/tex].
5. Expressing as a decimal:
The exponent [tex]\( \frac{2}{3} \)[/tex] can be converted to its decimal form:
[tex]\[
\frac{2}{3} \approx 0.666666666666667
\][/tex]
Thus, the simplified expression in decimal powers is:
[tex]\[
x^{0.666666666666667}
\][/tex]
Therefore, the simplified form of the given expression [tex]\( \left(x^{\frac{2}{3}}\right)^{\frac{8}{8}} \)[/tex] is [tex]\( x^{0.666666666666667} \)[/tex].
