To solve the expression [tex]\(\left(3 x^2\right)^0\)[/tex] where [tex]\(x \neq 0\)[/tex], we need to follow some basic mathematical principles.
1. Understand the Exponent Zero Property:
Any non-zero number raised to the power of zero is equal to 1. This is a fundamental property of exponents which states:
[tex]\[
a^0 = 1 \quad \text{for any} \quad a \neq 0
\][/tex]
2. Apply the Exponent Rule to the Given Expression:
In the expression [tex]\(\left(3 x^2\right)^0\)[/tex], the base is [tex]\(3 x^2\)[/tex]. According to the property of exponents:
[tex]\[
\left(3 x^2\right)^0 = 1 \quad \text{for any} \quad 3 x^2 \neq 0
\][/tex]
3. Consider the Condition [tex]\(x \neq 0\)[/tex]:
The given condition [tex]\(x \neq 0\)[/tex] ensures that [tex]\(3 x^2 \neq 0\)[/tex]. This means [tex]\(3 x^2\)[/tex] is always a non-zero number when [tex]\(x\)[/tex] is non-zero.
4. Conclusion:
Since [tex]\(3 x^2\)[/tex] is a non-zero number and any non-zero number raised to the power of zero equals 1, we have:
[tex]\[
\left(3 x^2\right)^0 = 1
\][/tex]
Thus, the expression [tex]\(\left(3 x^2\right)^0\)[/tex] simplifies to [tex]\(1\)[/tex] for any non-zero [tex]\(x\)[/tex]. Therefore:
[tex]\[
\boxed{1}
\][/tex]
