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Answer :
To solve the equation [tex]\(\left(\frac{1}{5}\right)^x = 625\)[/tex], let's follow these steps:
1. Express 625 as a Power of 5:
We know that [tex]\(625\)[/tex] can be expressed as a power of 5. Calculating, we find:
[tex]\[
625 = 5 \times 5 \times 5 \times 5 = 5^4
\][/tex]
So, the equation becomes:
[tex]\[
\left(\frac{1}{5}\right)^x = 5^4
\][/tex]
2. Rewrite [tex]\(\left(\frac{1}{5}\right)^x\)[/tex] as a Power of 5:
The expression [tex]\(\left(\frac{1}{5}\right)^x\)[/tex] can be rewritten using the negative exponent rule:
[tex]\[
\left(\frac{1}{5}\right)^x = (5^{-1})^x = 5^{-x}
\][/tex]
3. Set the Exponents Equal:
Since the bases are the same on both sides (both are base 5), we can set the exponents equal to each other:
[tex]\[
-x = 4
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
To find [tex]\(x\)[/tex], solve the equation:
[tex]\[
-x = 4
\][/tex]
Multiply both sides by [tex]\(-1\)[/tex] to solve for [tex]\(x\)[/tex]:
[tex]\[
x = -4
\][/tex]
So, the solution is [tex]\(x = -4\)[/tex].
1. Express 625 as a Power of 5:
We know that [tex]\(625\)[/tex] can be expressed as a power of 5. Calculating, we find:
[tex]\[
625 = 5 \times 5 \times 5 \times 5 = 5^4
\][/tex]
So, the equation becomes:
[tex]\[
\left(\frac{1}{5}\right)^x = 5^4
\][/tex]
2. Rewrite [tex]\(\left(\frac{1}{5}\right)^x\)[/tex] as a Power of 5:
The expression [tex]\(\left(\frac{1}{5}\right)^x\)[/tex] can be rewritten using the negative exponent rule:
[tex]\[
\left(\frac{1}{5}\right)^x = (5^{-1})^x = 5^{-x}
\][/tex]
3. Set the Exponents Equal:
Since the bases are the same on both sides (both are base 5), we can set the exponents equal to each other:
[tex]\[
-x = 4
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
To find [tex]\(x\)[/tex], solve the equation:
[tex]\[
-x = 4
\][/tex]
Multiply both sides by [tex]\(-1\)[/tex] to solve for [tex]\(x\)[/tex]:
[tex]\[
x = -4
\][/tex]
So, the solution is [tex]\(x = -4\)[/tex].
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Rewritten by : Jeany
