Question

if x+1/x=3,then the value of x5+1/x5 is

Answer 1

Step 1: Given data

$x+\frac{1}{x}=3---(1)$

$x^{5} +\frac{1}{x^{5} }=?$

Step 2: Calculating the required value

squaring both sides of $(1)$,

$(x+\frac{1}{x})^{2} =3^{2} \\\\x^{2} +\frac{1}{x^{2} } +2=9\\\\x^{2} +\frac{1}{x^{2} } =7---(2)$

taking cube on both sides of  $(1)$,

$(x+\frac{1}{x})^{3} =3^{3} \\\\x^{3} +\frac{1}{x^{3} } +3\times x\times\frac{1}{x}(x+\frac{1}{x}) =27\\\\x^{3} +\frac{1}{x^{3} } +3(3) =27\\\\\\x^{3} +\frac{1}{x^{3} } +9 =27\\\\x^{3} +\frac{1}{x^{3} } =27-9=18----(3)$

squaring both sides of $(2)$,

$(x^{2} +\frac{1}{x^{2} })^{2} =7^{2} \\\\x^{4} +\frac{1}{x^{4} } +2=49\\\\x^{4} +\frac{1}{x^{4} }=49-2=47---(4)$

Multiplying $(1)$ by $(4)$,

$(x+\frac{1}{x})(x^{4} +\frac{1}{x^{4} })=3\times47\\\\x^{5} +\frac{1}{x^{3} }+x^{3} +\frac{1}{x^{5} }=141\\\\x^{5} +\frac{1}{x^{5} }=141-(\frac{1}{x^{3} }+x^{3})\\\\x^{5} +\frac{1}{x^{5} }=141-(18)=123$

Hence, the value of the given expression is $123$.

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Answer 2

Answer:

The value of $x^5+\frac{1}{x^5} = 123$

Step-by-step explanation:

• Now it is given to us that
  $x+\frac{1}{x} = 3$ ----1
• When we square on both the sides we get
  $(x + \frac{1}{x} )^2 = 3^2\\x^2 + \frac{1}{x^2} +2 = 9\\x^2 + \frac{1}{x^2} = 7$-------- 2
• Now again squaring on both the sides we get
  $(x^2 + \frac{1}{x^2})^2 = 7^2\\x^4 + \frac{1}{x^4} +2 = 49\\x^4 + \frac{1}{x^4} = 47$---------- 3
• Now by taking cube on both the sides in equation 1 we get
  $(x + \frac{1}{x} )^3 = 3^3\\x^3 + \frac{1}{x^3} + 3 (x + \frac{1}{x}) = 27\\x^3 + \frac{1}{x^3} +3*3 = 27\\x^3 + \frac{1}{x^3} = 18$------------- 4
• Now on multiplying equation 3 with equation 1 we get
  $(x+\frac{1}{x}) (x^4+\frac{1}{x^4}) = 3 * 47\\x^5+ \frac{1}{x^5}+ x^3+ \frac{1}{x^3} = 141\\x^5+ \frac{1}{x^5} + 18 =141\\x^5+ \frac{1}{x^5} =141-18\\x^5+ \frac{1}{x^5} = 123$
• So the value of $x^5+\frac{1}{x^5} = 123$
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