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Determine which expression is not equivalent to \( 5^{5 m^{2}} \cdot 5^{5 m} \) \( 5^{5 m^{2}+5 m} \) \( 25^{\frac{5}{2} m^{2}} \cdot 25^{\frac{5}{2} m} \) \( 5^{m^{2} \cdot 5 m} \) \( 5^{5\left(m^{2}+m\right)} \)

Ask by Floyd Whittaker. in the United States
Jan 25,2025

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Tutor-Verified Answer

Answer

\( 5^{m^{2} \cdot 5 m} \) is not equivalent to \( 5^{5 m^{2}} \cdot 5^{5 m} \).

Solution

To determine which expression is **not** equivalent to \( 5^{5 m^{2}} \cdot 5^{5 m} \), let's analyze each option step-by-step. ### Original Expression: \[ 5^{5 m^{2}} \cdot 5^{5 m} \] Using the **laws of exponents**, specifically \( a^b \cdot a^c = a^{b + c} \), we can combine the terms: \[ 5^{5 m^{2} + 5 m} = 5^{5(m^{2} + m)} \] ### Option 1: \[ 5^{5 m^{2} + 5 m} \] - This is directly equivalent to the original expression. ### Option 2: \[ 25^{\frac{5}{2} m^{2}} \cdot 25^{\frac{5}{2} m} \] First, recognize that \( 25 = 5^2 \): \[ 25^{\frac{5}{2} m^{2}} = (5^2)^{\frac{5}{2} m^{2}} = 5^{5 m^{2}} \] \[ 25^{\frac{5}{2} m} = (5^2)^{\frac{5}{2} m} = 5^{5 m} \] Now, multiply them: \[ 5^{5 m^{2}} \cdot 5^{5 m} = 5^{5 m^{2} + 5 m} \] - This is equivalent to the original expression. ### Option 3: \[ 5^{m^{2} \cdot 5 m} = 5^{5 m^{3}} \] - This exponent simplifies to \( 5 m^{3} \), which does **not** equal \( 5 m^{2} + 5 m \) except for specific values of \( m \). Hence, this expression is **not** equivalent to the original. ### Option 4: \[ 5^{5(m^{2} + m)} \] - This simplifies to \( 5^{5 m^{2} + 5 m} \), which is equivalent to the original expression. ### Conclusion: **Option 3:** \( 5^{m^{2} \cdot 5 m} \) is **not equivalent** to \( 5^{5 m^{2}} \cdot 5^{5 m} \). **Answer:** \( 5^{m^{2} \cdot 5 m} \) is not equivalent to \( 5^{5 m^{2}} \cdot 5^{5 m} \)

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The Deep Dive

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 \( 5^{m^{2} \cdot 5 m} \).

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