Question

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determine whether \\(\frac{25^m}{5}\\) is equivalent to each of the following expressions.
equivalent not equivalent
\\(25^{2m - 1}\\) \\(\circ\\) \\(\circ\\)
\\(5^{2m - 1}\\) \\(\circ\\) \\(\circ\\)
\\(25^{m - 1}\\) \\(\circ\\) \\(\circ\\)
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Explanation:

Step1: Simplify the original expression

First, rewrite $25$ as $5^2$, so $25^m = (5^2)^m = 5^{2m}$. Then divide by $5$:
$\frac{25^m}{5} = \frac{5^{2m}}{5^1} = 5^{2m-1}$
Also, rewrite $5^{2m-1}$ as $5^{2(m-1)+1} = 25^{m-1}$ (since $5^{2(m-1)} = 25^{m-1}$, then multiply by $5^1$ is not needed—wait, correct rewrite: $\frac{25^m}{25} = 25^{m-1}$, and $\frac{25^m}{5} = 25^m \times 5^{-1} = (5^2)^m \times 5^{-1} = 5^{2m-1}$, and $25^{m-1} = 5^{2(m-1)} = 5^{2m-2}$, no—wait correct step:
$\frac{25^m}{5} = 25^m \times 5^{-1} = (5^2)^m \times 5^{-1} = 5^{2m} \times 5^{-1} = 5^{2m-1}$
Now rewrite $5^{2m-1}$ in terms of 25: $5^{2m-1} = \frac{5^{2m}}{5} = \frac{(5^2)^m}{5} = \frac{25^m}{5}$, and $25^{m-1} = \frac{25^m}{25} = \frac{25^m}{5^2}$, which is not equal to $\frac{25^m}{5}$. $25^{2m-1} = 25^{2m} \times 25^{-1} = (25^m)^2 \times \frac{1}{25}$, which is not equal to $\frac{25^m}{5}$.

Step2: Compare with first expression $25^{2m-1}$

$\frac{25^m}{5}
eq 25^{2m-1}$, since $25^{2m-1} = \frac{(25^m)^2}{25}$, which is a different expression.

Step3: Compare with second expression $5^{2m-1}$

We already simplified $\frac{25^m}{5}$ to $5^{2m-1}$, so they are equivalent.

Step4: Compare with third expression $25^{m-1}$

$25^{m-1} = \frac{25^m}{25}$, while $\frac{25^m}{5} = 5 \times 25^{m-1}$, so they are not equivalent.

Answer:

• $25^{2m-1}$: Not equivalent
• $5^{2m-1}$: Equivalent
• $25^{m-1}$: Not equivalent