To simplify the expression [tex]\(\left(c^2\right)^{-3}\)[/tex], we will use the properties of exponents. Let's go through the steps together.
1. Understand the notation:
[tex]\(\left(c^2\right)^{-3}\)[/tex] means raising [tex]\(c^2\)[/tex] to the power of [tex]\(-3\)[/tex].
2. Use the power of a power rule:
The property [tex]\((a^m)^n = a^{m \cdot n}\)[/tex] allows us to simplify expressions where one exponent is raised to another. Applying this rule:
[tex]\[
\left(c^2\right)^{-3} = c^{2 \cdot (-3)}
\][/tex]
3. Multiply the exponents:
Compute the product of the exponents [tex]\(2 \cdot (-3)\)[/tex]:
[tex]\[
2 \cdot (-3) = -6
\][/tex]
4. Rewrite the expression:
So, we have:
[tex]\[
c^{2 \cdot (-3)} = c^{-6}
\][/tex]
5. Simplify the negative exponent:
An expression with a negative exponent [tex]\(a^{-n}\)[/tex] can be rewritten as [tex]\(\frac{1}{a^n}\)[/tex]. Therefore, we translate [tex]\(c^{-6}\)[/tex] to:
[tex]\[
c^{-6} = \frac{1}{c^6}
\][/tex]
So, the simplified form of [tex]\(\left(c^2\right)^{-3}\)[/tex] is [tex]\(\frac{1}{c^6}\)[/tex].
Hence, among the given options, the correct answer is [tex]\(\boxed{\frac{1}{c^6}}\)[/tex].
