Respuesta :
123 is the value of [tex]x^{5} + \frac{1}{x^{5} }[/tex] when the value of x + [tex]\frac{1}{x}[/tex] is 3.
Which algebraic identities are used here?
The algebraic equations are those which are valid for all values of variables in them are called algebraic identities.
Here we used two algebraic identities:
[tex](x + y)^{2} = x^{2} + y^{2} + 2xy\\(x + y)^{3} = x^{3} + y^{3} + 3x^{2} y + 3 y^{2} x[/tex]
Given that,
x + [tex]\frac{1}{x}[/tex] = 3
Take square of both sides:
[tex]x^{2} + \frac{1}{x^{2} } + 2 = 9[/tex]
or
[tex]x^{2} + \frac{1}{x^{2} } = 9 -2 = 7[/tex]
Similarly,
x + [tex]\frac{1}{x}[/tex] = 3
Take cube of both sides:
[tex]x^{3} + \frac{x}{x^{3} } + 9 = 27 \\x^{3} + \frac{x}{x^{3} } = 27 - 9 = 18 \\[/tex]
Multiply cubic and square equations, we get:
[tex]x^{5} + \frac{1}{x^{5} }[/tex] + 3 = 18 x 17 = 126
[tex]x^{5} + \frac{1}{x^{5} }[/tex] = 126 - 3 = 123
∴ [tex]x^{5} + \frac{1}{x^{5} }[/tex] = 123
Hence the expression [tex]x^{5} + \frac{1}{x^{5} }\\[/tex] value is 123.
(Note the expression asked is different by minus sign, but generally these questions solved as above. It would be a printing mistake)
To learn more about algebraic identities, refer to the link:
https://brainly.com/question/953809
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