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Answer :
To solve the equation [tex]\( 5^x = \frac{1}{25} \)[/tex], let's go through the steps:
1. Rewrite the Fraction:
The fraction [tex]\(\frac{1}{25}\)[/tex] can be rewritten using an exponent. Notice that [tex]\(25\)[/tex] is the same as [tex]\(5^2\)[/tex]. Thus, [tex]\(\frac{1}{25}\)[/tex] can be written as [tex]\(5^{-2}\)[/tex] because a negative exponent indicates the reciprocal:
[tex]\[
\frac{1}{25} = \frac{1}{5^2} = 5^{-2}
\][/tex]
2. Equate the Exponents:
Now the equation becomes:
[tex]\[
5^x = 5^{-2}
\][/tex]
Since the bases are the same (both are 5), we can equate the exponents. This gives us:
[tex]\[
x = -2
\][/tex]
Therefore, the solution to the equation is [tex]\( x = -2 \)[/tex].
1. Rewrite the Fraction:
The fraction [tex]\(\frac{1}{25}\)[/tex] can be rewritten using an exponent. Notice that [tex]\(25\)[/tex] is the same as [tex]\(5^2\)[/tex]. Thus, [tex]\(\frac{1}{25}\)[/tex] can be written as [tex]\(5^{-2}\)[/tex] because a negative exponent indicates the reciprocal:
[tex]\[
\frac{1}{25} = \frac{1}{5^2} = 5^{-2}
\][/tex]
2. Equate the Exponents:
Now the equation becomes:
[tex]\[
5^x = 5^{-2}
\][/tex]
Since the bases are the same (both are 5), we can equate the exponents. This gives us:
[tex]\[
x = -2
\][/tex]
Therefore, the solution to the equation is [tex]\( x = -2 \)[/tex].
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