Question

if p/q+q/p=2 then what is the value of (p/q)^23+(q/p)^7​

Answer 1

Step-by-step explanation:

(p/q) +(q/p) =2

=>p^2+q^2=2pq

=>p^2-2pq+q^2=0

=>(p-q) ^2=0

=>p-q=0

=>p=q

=>p/q=1 or. q/p=1

Now,(p/q)^23+(q/p)^10

=1^23+1^10

=1+1

=2

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Answer 2

Given:

• $\frac{p}{q}+\frac{q}{p} =2$

To Find:

• The value of $(\frac{p}{q})^{23} +(\frac{p}{q}) ^7$

Solution:

• Let us name $\frac{p}{q} = y$  and $\frac{q}{p} = \frac{1}{y}$  it will be easy while calculations.
• Consider, $y+\frac{1}{y} = 2$  
• ⇒ $\frac{y^2+1}{y} = 2$ ⇒ $y^2+1 = 2y$
• ⇒ $y^2-2y+1 = 0$ (quadratic equation)
• ⇒ y(y-1)-1(y-1) = 0
• (y-1) (y-1) = 0
• y-1 = 0 and y-1 = 0
• y = 1 and y = 1
• This implies that  $\frac{p}{q}$ = 1 and $\frac{q}{p}$ = 1
• Consider, $(\frac{p}{q})^{23} +(\frac{p}{q}) ^7$ Substitute the values in the given equation we get,
• ⇒$(\frac{p}{q})^{23} +(\frac{p}{q}) ^7$ =  $1^{23} +1 ^7 = 1+1 = 2$  

∴The value of $(\frac{p}{q})^{23} +(\frac{p}{q}) ^7$ = 2.