Question

If x power 8+1/ x power 8 equals to 47 then find the values of x power 4 + 1/xpower 4 and X ² +1/x²

Answer 1

Answer:

(x  

2

+  

x  

2

 

1

​

)  

2

=(x  

2

)  

2

+(  

x  

2

 

1

​

)  

2

+2(x  

2

)(  

x  

2

 

1

​

)

                              =x  

4

+  

x  

4

 

1

​

+2

                              =47+2

                              =49

=>x  

2

+  

x  

2

 

1

​

=7

=>(x+  

x

1

​

)  

2

=x  

2

+  

x  

2

 

1

​

+2(x)(  

x

1

​

)

                                =7+2

                                =9

=>x+  

x

1

​

=3

We know that,

x  

3

+  

x  

3

 

1

​

=(x+  

x

1

​

)(x  

2

+  

x  

2

 

1

​

−x×  

x

1

​

)

                              =3(7−1)

                              =18

Step-by-step explanation:

Answer 2

Answer:

$(x^{4} +\frac{1}{x^{4} } )$  = 7 and $(x^{2} +\frac{1}{x^{2} } )$ = 3

Step-by-step explanation:

Given :- It is given that $x^{8} +\frac{1}{x^{8} }$ = 47 .

To Find :- Value of  $(x^{4} +\frac{1}{x^{4} } )$  and  $(x^{2} +\frac{1}{x^{2} } )$ .

Solution :-

We know that,

                         ( a + b )² = a² + b² + 2 × a × b

Here it is given that,

$x^{8} +\frac{1}{x^{8} }$ = 47

We can write that,

$(x^{4} +\frac{1}{x^{4} } ) ^{2}$ = $(x^{4})^{2}$ + $(\frac{1}{x^{4}})^{2}$ + 2 ( $x^{4}$ ) ( $\frac{1}{x^{4} }$ )

⇒ $(x^{4} +\frac{1}{x^{4} } ) ^{2}$ = $x^{8}$ + $\frac{1}{x^{8} }$ + 2 × $x^{4}$ × $\frac{1}{x^{4} }$            

⇒ $(x^{4} +\frac{1}{x^{4} } ) ^{2}$  = 47 + 2

⇒ $(x^{4} +\frac{1}{x^{4} } ) ^{2}$  = 49

⇒ $(x^{4} +\frac{1}{x^{4} } )$  = √49

⇒ $(x^{4} +\frac{1}{x^{4} } )$  = 7

Now, we need to find $(x^{2} +\frac{1}{x^{2} } )$

$(x^{2} +\frac{1}{x^{2} } )^{2}$  =  $(x^{2}) ^{2}$ +  $(\frac{1}{x^{2} } )^{2}$ + 2 ( x² )( $\frac{1}{x^{2} }$ )

⇒ $(x^{2} +\frac{1}{x^{2} } )^{2}$ =  $x^{4}$ + $\frac{1}{x^{4}}$ + 2

⇒ $(x^{2} +\frac{1}{x^{2} } )^{2}$ = 7 + 2

⇒ $(x^{2} +\frac{1}{x^{2} } )^{2}$ = 9

⇒ $(x^{2} +\frac{1}{x^{2} } )$  = √9

⇒ $(x^{2} +\frac{1}{x^{2} } )$ = 3

Therefore, the value of  $(x^{4} +\frac{1}{x^{4} } )$  = 7 and  $(x^{2} +\frac{1}{x^{2} } )$ = 3 .

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