To find equivalent expressions to \(6 \cdot 6 \cdot 6 \cdot 6 \cdot 6\), we can simplify this expression first:
\[ 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 = 6^5 \]
Now, let's evaluate each choice:
Choice A: \((6^2)^3\)
Using the property of exponents \((a^m)^n = a^{mn}\):
\[ (6^2)^3 = 6^{2 \cdot 3} = 6^6 \]
This expression is not equivalent to \(6^5\).
Choice B: \(2^5 \cdot 3^5\)
We can express \(6\) as \(2 \cdot 3\):
\[ 6^5 = (2 \cdot 3)^5 = 2^5 \cdot 3^5 \]
This expression is equivalent to \(6^5\).
Choice C: \(\frac{6^6}{6^1}\)
Using the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\):
\[ \frac{6^6}{6^1} = 6^{6-1} = 6^5 \]
This expression is equivalent to \(6^5\).
Choice D: \(3^2 \cdot 2^3\)
Calculating this out:
\[ 3^2 \cdot 2^3 = 9 \cdot 8 = 72 \]
And calculating \(6^5\):
\[ 6^5 = 7776 \]
This expression is not equivalent to \(6^5\).
The two correct choices that are equivalent to \(6 \cdot 6 \cdot 6 \cdot 6 \cdot 6\) are:
- Choice B: \(2^5 \cdot 3^5\)
- Choice C: \(\frac{6^6}{6^1}\)
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