Sure! Let's simplify the expression [tex]\(\left(2 y^3 \right)^{-3}\)[/tex] step by step.
1. Understand the Expression:
We need to simplify the expression [tex]\(\left(2 y^3 \right)^{-3}\)[/tex].
2. Apply the Power Rule:
The power rule states that [tex]\((a \cdot b)^n = a^n \cdot b^n\)[/tex]. Here, [tex]\(a = 2\)[/tex] and [tex]\(b = y^3\)[/tex], and [tex]\(n = -3\)[/tex].
So the expression becomes:
[tex]\[
(2 y^3)^{-3} = 2^{-3} \cdot (y^3)^{-3}
\][/tex]
3. Simplify Each Part:
- For [tex]\(2^{-3}\)[/tex]:
[tex]\[
2^{-3} = \frac{1}{2^3} = \frac{1}{8}
\][/tex]
- For [tex]\((y^3)^{-3}\)[/tex]:
Using the power of a power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex],
[tex]\[
(y^3)^{-3} = y^{3 \cdot (-3)} = y^{-9}
\][/tex]
Rewrite [tex]\(y^{-9}\)[/tex] as [tex]\(\frac{1}{y^9}\)[/tex].
4. Combine the Results:
Now combine the simplified parts:
[tex]\[
2^{-3} \cdot (y^3)^{-3} = \frac{1}{8} \cdot \frac{1}{y^9} = \frac{1}{8 y^9}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\left(2 y^3\right)^{-3} = \frac{1}{8 y^9}
\][/tex]
This is the final simplified form of the given expression.
