To simplify the expression \( 54 \cdot 55^{4} \cdot 5^{5} \), we will first express 54 in terms of its prime factors and then combine like terms.
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Factor 54: \[ 54 = 2 \cdot 3^3 \]
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Express \( 55^{4} \): \[ 55 = 5 \cdot 11 \quad \Rightarrow \quad 55^{4} = (5 \cdot 11)^{4} = 5^{4} \cdot 11^{4} \]
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Reorganize the expression: Now we can rewrite the entire expression: \[ 54 \cdot 55^{4} \cdot 5^{5} = (2 \cdot 3^3) \cdot (5^{4} \cdot 11^{4}) \cdot (5^{5}) \]
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Combine like terms (the \( 5 \)'s): Combine \( 5^{4} \) and \( 5^{5} \): \[ 5^{4} \cdot 5^{5} = 5^{4 + 5} = 5^{9} \]
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Write the full expression in simpler terms: So now we have: \[ 2 \cdot 3^3 \cdot 5^{9} \cdot 11^{4} \]
At this stage, we could write the factors as separate terms, but the format you asked for is exponential form.
- Final expression: Therefore, the simplified expression can be written as: \[ 2^{1} \cdot 3^{3} \cdot 5^{9} \cdot 11^{4} \]
So the answer in exponential form is: \[ \boxed{2^{1} \cdot 3^{3} \cdot 5^{9} \cdot 11^{4}} \]
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