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21 Feb 2023, 05:14
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C
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Difficulty:
45%
(medium)
Question Stats:
62% (01:48) correct
38%
(02:30)
wrong
based on 13
sessions
History
Date
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Not Attempted Yet
If \(9^x + 9^{-x} = 62\), then what is the value of \(3^x + 3^{-x}\)?
A. \(\frac{1}{64}\)
B. \(\frac{1}{8}\)
C. \(8\)
D. \(62\)
E. \(64\)
A. \(\frac{1}{64}\)
B. \(\frac{1}{8}\)
C. \(8\)
D. \(62\)
E. \(64\)
21 Feb 2023, 05:15
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Official Solution:
If \(9^x + 9^{-x} = 62\), then what is the value of \(3^x + 3^{-x}\)?
A. \(\frac{1}{64}\)
B. \(\frac{1}{8}\)
C. \(8\)
D. \(62\)
E. \(64\)
We are given that \(9^x + 9^{-x} = 62\). To find the value of \(3^x + 3^{-x}\), we can square it and simplify:
\((3^x + 3^{-x})^2 = \)
\(=(3^x)^2 + 2*3^x*3^{-x} +(3^{-x})^2=\)
\(=9^x + 2*3^x*\frac{1}{3^{x}} +9^{-x}=\)
\(=9^x + 2+ 9^{-x}\)
\(=62 + 2 = 64\)
Now, since \(3^x + 3^{-x}\) squared is 64, then \(3^x + 3^{-x}\) is 8 (notice that it cannot be -8 since \(3^x + 3^{-x}\) is positivefor all values of \(x\)).
Answer: C
If \(9^x + 9^{-x} = 62\), then what is the value of \(3^x + 3^{-x}\)?
A. \(\frac{1}{64}\)
B. \(\frac{1}{8}\)
C. \(8\)
D. \(62\)
E. \(64\)
We are given that \(9^x + 9^{-x} = 62\). To find the value of \(3^x + 3^{-x}\), we can square it and simplify:
\((3^x + 3^{-x})^2 = \)
\(=(3^x)^2 + 2*3^x*3^{-x} +(3^{-x})^2=\)
\(=9^x + 2*3^x*\frac{1}{3^{x}} +9^{-x}=\)
\(=9^x + 2+ 9^{-x}\)
\(=62 + 2 = 64\)
Now, since \(3^x + 3^{-x}\) squared is 64, then \(3^x + 3^{-x}\) is 8 (notice that it cannot be -8 since \(3^x + 3^{-x}\) is positivefor all values of \(x\)).
Answer: C
01 Sep 2025, 15:32
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I don’t quite agree with the solution. How do you know to square the second equation? solution makes sense but would like to understand how you know to do that right away
