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Answer :
To solve the equation [tex]\(\left(1 \frac{1}{25}\right)^x-\left(\frac{1}{25}\right)^{3 / 4}=0\)[/tex], let's work through it step by step.
1. Understand the Equation: The equation involves two expressions being subtracted:
[tex]\[
\left(1 \frac{1}{25}\right)^x - \left(\frac{1}{25}\right)^{3/4} = 0
\][/tex]
This can be rewritten as:
[tex]\[
\left(\frac{26}{25}\right)^x = \left(\frac{1}{25}\right)^{3/4}
\][/tex]
2. Simplify Right Side: The right side [tex]\(\left(\frac{1}{25}\right)^{3/4}\)[/tex] can be interpreted as:
[tex]\[
\left(\frac{1}{25}\right)^{3/4} = \left(25^{-1}\right)^{3/4} = 25^{-3/4}
\][/tex]
3. Solve for [tex]\(x\)[/tex]: We have:
[tex]\[
\left(\frac{26}{25}\right)^x = 25^{-3/4}
\][/tex]
To solve for [tex]\(x\)[/tex], we'll equate the exponents by expressing both sides with a common base. The left side uses the base [tex]\(\frac{26}{25}\)[/tex], and the right side is a power of 25. This suggests we could express the left side using log properties or consider trial and error with matching powers.
4. Matching Exponents: For a general solution, equate the exponents assuming both sides can be simplified or directly evaluated:
[tex]\[
\log_{(26/25)} = -\frac{3}{4} \cdot \log_{(25)}
\][/tex]
5. Calculate [tex]\(x\)[/tex]: Solving this step gives the solution [tex]\(x\)[/tex] approximately:
[tex]\[
x \approx -61.553
\][/tex]
Thus, the solution is [tex]\(x = -61.553\)[/tex]. This means that raising [tex]\(\frac{26}{25}\)[/tex] to the power of [tex]\(-61.553\)[/tex] will give you the same numerical value as [tex]\(\left(\frac{1}{25}\right)^{3/4}\)[/tex].
1. Understand the Equation: The equation involves two expressions being subtracted:
[tex]\[
\left(1 \frac{1}{25}\right)^x - \left(\frac{1}{25}\right)^{3/4} = 0
\][/tex]
This can be rewritten as:
[tex]\[
\left(\frac{26}{25}\right)^x = \left(\frac{1}{25}\right)^{3/4}
\][/tex]
2. Simplify Right Side: The right side [tex]\(\left(\frac{1}{25}\right)^{3/4}\)[/tex] can be interpreted as:
[tex]\[
\left(\frac{1}{25}\right)^{3/4} = \left(25^{-1}\right)^{3/4} = 25^{-3/4}
\][/tex]
3. Solve for [tex]\(x\)[/tex]: We have:
[tex]\[
\left(\frac{26}{25}\right)^x = 25^{-3/4}
\][/tex]
To solve for [tex]\(x\)[/tex], we'll equate the exponents by expressing both sides with a common base. The left side uses the base [tex]\(\frac{26}{25}\)[/tex], and the right side is a power of 25. This suggests we could express the left side using log properties or consider trial and error with matching powers.
4. Matching Exponents: For a general solution, equate the exponents assuming both sides can be simplified or directly evaluated:
[tex]\[
\log_{(26/25)} = -\frac{3}{4} \cdot \log_{(25)}
\][/tex]
5. Calculate [tex]\(x\)[/tex]: Solving this step gives the solution [tex]\(x\)[/tex] approximately:
[tex]\[
x \approx -61.553
\][/tex]
Thus, the solution is [tex]\(x = -61.553\)[/tex]. This means that raising [tex]\(\frac{26}{25}\)[/tex] to the power of [tex]\(-61.553\)[/tex] will give you the same numerical value as [tex]\(\left(\frac{1}{25}\right)^{3/4}\)[/tex].
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