Let's simplify the expression [tex]\(\left(\frac{b^0}{a^3}\right)^{\frac{1}{3}}\)[/tex] step-by-step.
1. Understanding [tex]\(b^0\)[/tex]:
Any non-zero number raised to the power of 0 is equal to 1. Hence,
[tex]\[
b^0 = 1
\][/tex]
2. Rewrite the expression:
Substitute [tex]\(b^0\)[/tex] with 1 in the original expression:
[tex]\[
\left(\frac{1}{a^3}\right)^{\frac{1}{3}}
\][/tex]
3. Apply the exponent rule:
The rule [tex]\((x^m)^n = x^{mn}\)[/tex] can be applied here to further simplify:
[tex]\[
\left(\frac{1}{a^3}\right)^{\frac{1}{3}} = \left(\frac{1}{a^3}\right)^{1/3} = \left(1\right)^{1/3} \cdot \left(\frac{1}{a^3}\right)^{1/3}
\][/tex]
Now, analyze each part:
- [tex]\(\left(1\right)^{1/3} = 1\)[/tex]
- [tex]\(\left(\frac{1}{a^3}\right)^{1/3}\)[/tex]
4. Simplify [tex]\( \left(\frac{1}{a^3}\right)^{\frac{1}{3}} \)[/tex]:
Applying the property [tex]\( \left(\frac{1}{x}\right)^n = x^{-n}\)[/tex], we have:
[tex]\[
\left(\frac{1}{a^3}\right)^{1/3} = a^{-3 \cdot 1/3} = a^{-1}
\][/tex]
5. Final Expression:
Putting it all together:
[tex]\[
\left(\frac{b^0}{a^3}\right)^{\frac{1}{3}} = a^{-1}
\][/tex]
Therefore,
[tex]\[
a^{-1} = \frac{1}{a}
\][/tex]
The correct simplified form of the expression is [tex]\(\frac{1}{a}\)[/tex].
So, the correct answer is:
None of the given answers appear to be correct. This problem may contain a typo, or there may be a mistake in the given options. However, based on the provided options, the correct simplified form corresponding to our calculation would have been:
[tex]\[ \frac{1}{a} \][/tex]
