Determine which expression is not equivalent to \( 5^{5 m^{2}} \cdot 5^{5 m} \) \( 5^{5 m^{2}+5 m} \) \( 25^{\frac{5}{2} m^{2}} \cdot 25^{\frac{5}{2} m} \) \( 5^{m^{2} \cdot 5 m} \) \( 5^{5\left(m^{2}+m\right)} \)
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To determine which expression is not equivalent to \( 5^{5 m^{2}} \cdot 5^{5 m} \), we can simplify the given expression first. Using the property of exponents that states \( a^m \cdot a^n = a^{m+n} \), we have: \[ 5^{5 m^{2}} \cdot 5^{5 m} = 5^{5 m^{2} + 5 m} = 5^{5(m^{2} + m)} \] Now, let's analyze the other options one by one: 1. \( 5^{5 m^{2}+5 m} \) is equivalent since it simplifies to the same exponent. 2. \( 25^{\frac{5}{2} m^{2}} \cdot 25^{\frac{5}{2} m} \) can be rewritten as \( 5^{2(\frac{5}{2} m^{2})} \cdot 5^{2(\frac{5}{2} m)} = 5^{5 m^{2}} \cdot 5^{5 m} = 5^{5(m^{2} + m)} \), so this is also equivalent. 3. \( 5^{m^{2} \cdot 5 m} \) simplifies incorrectly because \( m^{2} \cdot 5 m \) is not equivalent to \( 5(m^{2} + m) \). This expression is incorrect. 4. \( 5^{5 \left(m^{2}+m\right)} \) is equivalent as it directly represents the same exponent. Thus, the expression that is not equivalent to \( 5^{5 m^{2}} \cdot 5^{5 m} \) is \( 5^{m^{2} \cdot 5 m} \).