To determine which expressions are equivalent to [tex]\( 25^x \)[/tex], we need to express [tex]\( 25^x \)[/tex] and the given options in a form that makes it easy to compare.
First, note that:
[tex]\[ 25 = 5^2 \][/tex]
Thus:
[tex]\[ 25^x = (5^2)^x \][/tex]
Using the properties of exponents, we know:
[tex]\[ (5^2)^x = 5^{2x} \][/tex]
So, the expression [tex]\( 25^x \)[/tex] can be rewritten as [tex]\( 5^{2x} \)[/tex].
Now, let's examine each given option:
Option A: [tex]\((5 \cdot 5)^x\)[/tex]
[tex]\[ (5 \cdot 5)^x = 5^x \cdot 5^x = 5^{x+x} = 5^{2x} \][/tex]
This matches [tex]\( 5^{2x} \)[/tex].
Option B: [tex]\(5 \cdot 5^{2x}\)[/tex]
[tex]\[ 5 \cdot 5^{2x} = 5^1 \cdot 5^{2x} = 5^{1 + 2x} \][/tex]
This does not match [tex]\( 5^{2x} \)[/tex].
Option C: [tex]\(5 \cdot 5^x\)[/tex]
[tex]\[ 5 \cdot 5^x = 5^1 \cdot 5^x = 5^{1 + x} \][/tex]
This does not match [tex]\( 5^{2x} \)[/tex].
Option D: [tex]\(5^{2x}\)[/tex]
This exactly matches [tex]\( 5^{2x} \)[/tex].
Option E: [tex]\(5^x \cdot 5^x\)[/tex]
[tex]\[ 5^x \cdot 5^x = 5^{x+x} = 5^{2x} \][/tex]
This matches [tex]\( 5^{2x} \)[/tex].
Option F: [tex]\(5^2 \cdot 5^x\)[/tex]
[tex]\[ 5^2 \cdot 5^x = 5^{2} \cdot 5^x = 5^{2 + x} \][/tex]
This does not match [tex]\( 5^{2x} \)[/tex].
In summary, the expressions that are equivalent to [tex]\( 25^x \)[/tex] are:
[tex]\[ \boxed{A, D, E} \][/tex]
