To determine which expression is equivalent to [tex]\(6^4 \cdot 6^3\)[/tex], we need to use the laws of exponents. When multiplying expressions with the same base, you add their exponents:
[tex]\[
6^4 \cdot 6^3 = 6^{4+3} = 6^7
\][/tex]
Now, we need to analyze each given expression to see which one simplifies correctly to [tex]\(6^7\)[/tex]:
1. [tex]\((6 \cdot 6 \cdot 6 \cdot 6) + (6 \cdot 6 \cdot 6)\)[/tex]
- This represents [tex]\(6^4 + 6^3\)[/tex], not [tex]\(6^4 \cdot 6^3\)[/tex]. This expression will not simplify to [tex]\(6^7\)[/tex].
2. [tex]\((6 + 6 + 6 + 6) \cdot (6 + 6 + 6)\)[/tex]
- This is the product of sums: [tex]\((6 \cdot 4) \cdot (6 \cdot 3)\)[/tex], which simplifies to [tex]\(24 \cdot 18\)[/tex]. This product does not give us [tex]\(6^7\)[/tex].
3. [tex]\((6 \cdot 6 \cdot 6 \cdot 6) \cdot (6 \cdot 6 \cdot 6)\)[/tex]
- This correctly represents [tex]\(6^4 \cdot 6^3\)[/tex], which simplifies to [tex]\(6^{4+3} = 6^7\)[/tex].
4. [tex]\((6 \cdot 6 \cdot 6 \cdot 6 \cdot 6) \cdot (6 \cdot 6 \cdot 6 \cdot 6)\)[/tex]
- This expression represents [tex]\(6^5 \cdot 6^4\)[/tex], which simplifies to [tex]\(6^{5+4} = 6^9\)[/tex]. This is not equivalent to [tex]\(6^4 \cdot 6^3\)[/tex].
Therefore, the expression that is equivalent to [tex]\(6^4 \cdot 6^3\)[/tex] is:
[tex]\( \boxed{(6 \cdot 6 \cdot 6 \cdot 6) \cdot (6 \cdot 6 \cdot 6)}\)[/tex]
The correct option is:
[tex]\(\boxed{3}\)[/tex]
