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Answer :
To solve the equation [tex]\(5^{-|x|} = \frac{1}{25}\)[/tex], follow these steps:
1. Understand the relationship: Notice that [tex]\(\frac{1}{25}\)[/tex] can be written as a power of 5. Since [tex]\(5^2 = 25\)[/tex], it follows that [tex]\(\frac{1}{25} = 5^{-2}\)[/tex].
2. Set the exponents equal: Because the bases (5) are the same on both sides of the equation, their exponents must be equal. Therefore, we can write:
[tex]\[
-|x| = -2
\][/tex]
3. Solve for [tex]\(|x|\)[/tex]: To solve for [tex]\(|x|\)[/tex], multiply both sides of the equation by -1:
[tex]\[
|x| = 2
\][/tex]
4. Solve the absolute value equation: An equation involving the absolute value [tex]\(|x| = 2\)[/tex] has two possible solutions, because [tex]\(x\)[/tex] can be either positive or negative:
- [tex]\(x = 2\)[/tex]
- [tex]\(x = -2\)[/tex]
Thus, the solutions to the equation [tex]\(5^{-|x|} = \frac{1}{25}\)[/tex] are [tex]\(x = 2\)[/tex] and [tex]\(x = -2\)[/tex].
1. Understand the relationship: Notice that [tex]\(\frac{1}{25}\)[/tex] can be written as a power of 5. Since [tex]\(5^2 = 25\)[/tex], it follows that [tex]\(\frac{1}{25} = 5^{-2}\)[/tex].
2. Set the exponents equal: Because the bases (5) are the same on both sides of the equation, their exponents must be equal. Therefore, we can write:
[tex]\[
-|x| = -2
\][/tex]
3. Solve for [tex]\(|x|\)[/tex]: To solve for [tex]\(|x|\)[/tex], multiply both sides of the equation by -1:
[tex]\[
|x| = 2
\][/tex]
4. Solve the absolute value equation: An equation involving the absolute value [tex]\(|x| = 2\)[/tex] has two possible solutions, because [tex]\(x\)[/tex] can be either positive or negative:
- [tex]\(x = 2\)[/tex]
- [tex]\(x = -2\)[/tex]
Thus, the solutions to the equation [tex]\(5^{-|x|} = \frac{1}{25}\)[/tex] are [tex]\(x = 2\)[/tex] and [tex]\(x = -2\)[/tex].
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