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Answer :
We start with the exponential equation:
[tex]$$
5^{5x} = \frac{1}{25}.
$$[/tex]
Notice that the number 25 can be written as [tex]$5^2$[/tex], so we can express [tex]$\frac{1}{25}$[/tex] as:
[tex]$$
\frac{1}{25} = \frac{1}{5^2} = 5^{-2}.
$$[/tex]
Now the equation becomes:
[tex]$$
5^{5x} = 5^{-2}.
$$[/tex]
Since the bases on both sides of the equation are the same (both are 5), we can set the exponents equal to each other:
[tex]$$
5x = -2.
$$[/tex]
To solve for [tex]$x$[/tex], divide both sides by 5:
[tex]$$
x = \frac{-2}{5}.
$$[/tex]
Converting this fraction to a decimal gives:
[tex]$$
x = -0.4.
$$[/tex]
Thus, the solution to the equation is:
[tex]$$
\boxed{-0.4}.
$$[/tex]
[tex]$$
5^{5x} = \frac{1}{25}.
$$[/tex]
Notice that the number 25 can be written as [tex]$5^2$[/tex], so we can express [tex]$\frac{1}{25}$[/tex] as:
[tex]$$
\frac{1}{25} = \frac{1}{5^2} = 5^{-2}.
$$[/tex]
Now the equation becomes:
[tex]$$
5^{5x} = 5^{-2}.
$$[/tex]
Since the bases on both sides of the equation are the same (both are 5), we can set the exponents equal to each other:
[tex]$$
5x = -2.
$$[/tex]
To solve for [tex]$x$[/tex], divide both sides by 5:
[tex]$$
x = \frac{-2}{5}.
$$[/tex]
Converting this fraction to a decimal gives:
[tex]$$
x = -0.4.
$$[/tex]
Thus, the solution to the equation is:
[tex]$$
\boxed{-0.4}.
$$[/tex]
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