Sure, let's simplify the expression [tex]\(\frac{12^6}{6^6}\)[/tex] step-by-step.
1. First, express [tex]\(12\)[/tex] and [tex]\(6\)[/tex] in terms of their prime factors:
- [tex]\(12\)[/tex] can be written as [tex]\(12 = 2^2 \cdot 3\)[/tex].
- [tex]\(6\)[/tex] can be written as [tex]\(6 = 2 \cdot 3\)[/tex].
2. Rewrite [tex]\(12^6\)[/tex] and [tex]\(6^6\)[/tex] using these prime factorizations:
- [tex]\(12^6 = (2^2 \cdot 3)^6 = (2^2)^6 \cdot 3^6 = 2^{12} \cdot 3^6\)[/tex].
- [tex]\(6^6 = (2 \cdot 3)^6 = 2^6 \cdot 3^6\)[/tex].
3. Now, divide [tex]\(12^6\)[/tex] by [tex]\(6^6\)[/tex]:
[tex]\[
\frac{12^6}{6^6} = \frac{2^{12} \cdot 3^6}{2^6 \cdot 3^6}
\][/tex]
4. Simplify the fraction by cancelling out the common factors in the numerator and the denominator:
[tex]\[
\frac{2^{12} \cdot 3^6}{2^6 \cdot 3^6} = \frac{2^{12}}{2^6} \cdot \frac{3^6}{3^6}
\][/tex]
[tex]\[
= 2^{12-6} \cdot 3^{6-6}
\][/tex]
[tex]\[
= 2^6 \cdot 3^0
\][/tex]
[tex]\[
= 2^6 \cdot 1
\][/tex]
[tex]\[
= 2^6
\][/tex]
5. Compare [tex]\(2^6\)[/tex] with the choices given:
- [tex]\(\frac{2^6}{2}\)[/tex] is not equivalent to [tex]\(2^6\)[/tex].
- [tex]\(\frac{1}{2^6}\)[/tex] is not equivalent to [tex]\(2^6\)[/tex].
- [tex]\(2^0 = 1\)[/tex] is not equivalent to [tex]\(2^6\)[/tex].
- [tex]\(2^3 \cdot 2^3 = 2^6\)[/tex], which is equivalent to [tex]\(2^6\)[/tex].
Therefore, the expression equivalent to [tex]\(\frac{12^6}{6^6}\)[/tex] is [tex]\(2^3 \cdot 2^3\)[/tex].
So, the correct choice is:
[tex]\(2^3 \cdot 2^3\)[/tex].
