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To find the value of [tex]\( x^4 + \frac{1}{x^4} \)[/tex] given that [tex]\( x + \frac{1}{x} = 9 \)[/tex], we will proceed step-by-step through algebraic manipulations.
1. Step 1: Find [tex]\( x^2 + \frac{1}{x^2} \)[/tex]
Start by squaring both sides of the given equation:
[tex]\[ \left( x + \frac{1}{x} \right)^2 = 9^2 \][/tex]
Expanding the left side using the binomial theorem:
[tex]\[ x^2 + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 81 \][/tex]
Simplify the expression:
[tex]\[ x^2 + 2 + \frac{1}{x^2} = 81 \][/tex]
Subtract 2 from both sides:
[tex]\[ x^2 + \frac{1}{x^2} = 79 \][/tex]
2. Step 2: Find [tex]\( x^4 + \frac{1}{x^4} \)[/tex]
Next, square the expression [tex]\( x^2 + \frac{1}{x^2} \)[/tex]:
[tex]\[ \left( x^2 + \frac{1}{x^2} \right)^2 = 79^2 \][/tex]
Expand the left side:
[tex]\[ x^4 + 2 \cdot x^2 \cdot \frac{1}{x^2} + \frac{1}{x^4} = 6241 \][/tex]
Simplify the expression:
[tex]\[ x^4 + 2 + \frac{1}{x^4} = 6241 \][/tex]
Subtract 2 from both sides:
[tex]\[ x^4 + \frac{1}{x^4} = 6239 \][/tex]
Thus, the value of [tex]\( x^4 + \frac{1}{x^4} \)[/tex] is
[tex]\[ \boxed{6239} \][/tex]
1. Step 1: Find [tex]\( x^2 + \frac{1}{x^2} \)[/tex]
Start by squaring both sides of the given equation:
[tex]\[ \left( x + \frac{1}{x} \right)^2 = 9^2 \][/tex]
Expanding the left side using the binomial theorem:
[tex]\[ x^2 + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 81 \][/tex]
Simplify the expression:
[tex]\[ x^2 + 2 + \frac{1}{x^2} = 81 \][/tex]
Subtract 2 from both sides:
[tex]\[ x^2 + \frac{1}{x^2} = 79 \][/tex]
2. Step 2: Find [tex]\( x^4 + \frac{1}{x^4} \)[/tex]
Next, square the expression [tex]\( x^2 + \frac{1}{x^2} \)[/tex]:
[tex]\[ \left( x^2 + \frac{1}{x^2} \right)^2 = 79^2 \][/tex]
Expand the left side:
[tex]\[ x^4 + 2 \cdot x^2 \cdot \frac{1}{x^2} + \frac{1}{x^4} = 6241 \][/tex]
Simplify the expression:
[tex]\[ x^4 + 2 + \frac{1}{x^4} = 6241 \][/tex]
Subtract 2 from both sides:
[tex]\[ x^4 + \frac{1}{x^4} = 6239 \][/tex]
Thus, the value of [tex]\( x^4 + \frac{1}{x^4} \)[/tex] is
[tex]\[ \boxed{6239} \][/tex]
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