To determine the expression equivalent to [tex]\(\left(2 w^{-2}\right)^3\left(8 w^6\right)\)[/tex], we need to simplify it step-by-step.
1. Start by expanding [tex]\(\left(2 w^{-2}\right)^3\)[/tex]:
- When raising a term to a power, we apply the exponent to both the coefficient and the variable separately.
- [tex]\((2 w^{-2})^3 = 2^3 \cdot (w^{-2})^3\)[/tex]
- [tex]\(2^3 = 8\)[/tex]
- [tex]\((w^{-2})^3 = w^{-6}\)[/tex]
Therefore, [tex]\(\left(2 w^{-2}\right)^3 = 8 w^{-6}\)[/tex].
2. Next, simplify [tex]\(\left(8 w^6\right)\)[/tex]:
- The term [tex]\(8 w^6\)[/tex] remains as it is, because it is already in its simplest form.
3. Multiply the results from steps 1 and 2:
[tex]\[
(8 w^{-6}) \cdot (8 w^6)
\][/tex]
4. Combine the coefficients:
- [tex]\(8 \cdot 8 = 64\)[/tex]
5. Combine the variables using the property of exponents that [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]:
- [tex]\(w^{-6} \cdot w^6 = w^{-6+6} = w^0\)[/tex]
- Since any number raised to the power of 0 is 1, [tex]\(w^0 = 1\)[/tex]
6. Therefore, we have:
[tex]\[
64 \cdot 1 = 64
\][/tex]
So, the expression simplifies to 64. The correct answer is:
B. 64
