Let's find the expression that is equivalent to [tex]\((-4 a b c)^3\)[/tex].
1. Identify the base expression and the exponent:
- The base expression is [tex]\(-4 a b c\)[/tex].
- The exponent is [tex]\(3\)[/tex].
2. Distribute the exponent to each factor in the base:
- The original base is a product of [tex]\(-4\)[/tex], [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex].
- When raising a product to a power, we can raise each factor to that power individually. Hence, the expression becomes:
[tex]\[
(-4 a b c)^3 = (-4)^3 \cdot (a)^3 \cdot (b)^3 \cdot (c)^3
\][/tex]
3. Calculate [tex]\((-4)^3\)[/tex]:
- We first compute the cube of [tex]\(-4\)[/tex]:
[tex]\[
(-4)^3 = -4 \times -4 \times -4
\][/tex]
- Multiplying out:
[tex]\[
(-4) \times (-4) = 16 \\
16 \times (-4) = -64
\][/tex]
Thus, [tex]\((-4)^3 = -64\)[/tex].
4. Combine the results with the variables raised to their respective powers:
- We now have:
[tex]\[
(-4 a b c)^3 = -64 \cdot a^3 \cdot b^3 \cdot c^3
\][/tex]
5. Write the final expression in standard form:
- The final form of the expression is:
[tex]\[
-64 a^3 b^3 c^3
\][/tex]
Now, we compare this result with the given options. The expression [tex]\((-4 a b c)^3\)[/tex] simplifies to [tex]\(-64 a^3 b^3 c^3\)[/tex].
Hence, the correct answer is:
[tex]\[
\boxed{-64 a^3 b^3 c^3}
\][/tex]
