To find the cube root of the expression [tex]\(-729a^9b^6\)[/tex], we will follow these steps:
1. Factor the expression into its components:
- The constant term is [tex]\(-729\)[/tex].
- The [tex]\(a\)[/tex] term is [tex]\(a^9\)[/tex].
- The [tex]\(b\)[/tex] term is [tex]\(b^6\)[/tex].
2. Find the cube root of the constant term:
- The cube root of [tex]\(-729\)[/tex] is [tex]\( -9 \)[/tex]. This is because [tex]\( (-9)^3 = -729 \)[/tex].
3. Find the cube root of the [tex]\(a\)[/tex] term:
- The given term is [tex]\( a^9 \)[/tex].
- The cube root of [tex]\( a^9 \)[/tex] is [tex]\( a^{9/3} = a^3 \)[/tex].
4. Find the cube root of the [tex]\(b\)[/tex] term:
- The given term is [tex]\( b^6 \)[/tex].
- The cube root of [tex]\( b^6 \)[/tex] is [tex]\( b^{6/3} = b^2 \)[/tex].
5. Combine the cube roots:
- Combining the results from steps 2, 3, and 4, we get:
[tex]\[ \text{Cube root of } -729a^9b^6 = -9 \cdot a^3 \cdot b^2 = -9a^3b^2 \][/tex]
So, the cube root of [tex]\(-729 a^9 b^6\)[/tex] is [tex]\(\boxed{-9a^3b^2}\)[/tex].
