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प्रश्न
If \[x + \frac{1}{x} = 9,\] find the value of \[x^4 + \frac{1}{x^4} .\]
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उत्तर
Let us consider the following equation: \[x + \frac{1}{x} = 9\] Squaring both sides, we get:
\[\left( x + \frac{1}{x} \right)^2 = \left( 9 \right)^2 = 81\]
\[ \Rightarrow \left( x + \frac{1}{x} \right)^2 = 81\]
\[ \Rightarrow x^2 + 2 \times x \times \frac{1}{x} + \left( \frac{1}{x} \right)^2 = 81\]
\[ \Rightarrow x^2 + 2 + \frac{1}{x^2} = 81\]
\[\Rightarrow x^2 + \frac{1}{x^2} = 79\] (Subtracting 2 from both sides)
Now, squaring both sides again, we get:
\[\left( x^2 + \frac{1}{x^2} \right)^2 = \left( 79 \right)^2 = 6241\]
\[ \Rightarrow \left( x^2 + \frac{1}{x^2} \right)^2 = 6241\]
\[ \Rightarrow \left( x^2 \right)^2 + 2\left( x^2 \right)\left( \frac{1}{x^2} \right) + \left( \frac{1}{x^2} \right)^2 = 6241\]
\[ \Rightarrow x^4 + 2 + \frac{1}{x^4} = 6241\]
\[\Rightarrow x^4 + \frac{1}{x^4} = 6239\]
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