Question

P+1/p=5 then prove
p³+1/p³=110

Answer 1

Given :-  

$\sf \bullet \;\; p + \dfrac{1}{p} = 5$

To Prove :-  

$\sf \bullet \;\; p^{3} + \dfrac{1}{p^{3}} = 110$

Proof :-  

~By cubing both sides of the given equation.  

$\sf : \; \implies \bigg\{ p + \dfrac{1}{p} \bigg\}^{3} = \{ 3 \}^{3}$

~As we know that,

       ( a + b ) = a³ + b³ + 3ab( a + b )

$\sf : \; \implies p^{3} + \dfrac{1}{p^{3}} + \bigg\{ 3 \times p \times \dfrac{1}{p} \bigg\} \times p + \dfrac{1}{p} = 125$

$\sf : \; \implies p^{3} + \dfrac{1}{p^{3}} + 3 \times \bigg\{ p + \dfrac{1}{p} \bigg\} = 125$

~Putting value of p + 1/p as 5 [ Given to us ]  

$\sf : \; \implies p^{3} + \dfrac{1}{p^{3}} + 3 \times 5 = 125$ 

$\sf : \; \implies p^{3} + \dfrac{1}{p^{3}} + 15 = 125$

 

$\sf : \; \implies p^{3} + \dfrac{1}{p^{3}} = 125 -15$

$\sf : \; \implies p^{3} + \dfrac{1}{p^{3}} = 110$

Hence, proved

Answer 2

Step-by-step explanation:

Given:-

p + 1/p = 5

To prove:-

p^3 + 1/p^3 = 110

Solution:-

We have,

p + 1/p = 5

On, cubing on both sides, we get

(p + 1/p)^3 = (5)^3

Now, applying algebraic Identity because our expression in the form of:

(a + b)^3 = a^ + b^³ + 3ab( a + b )

Where, we have to put in our expression a = p and b = 1/p , we get

→ (p)^3 + (1/p)^3 + 3(p)(1/p) (p + 1/p) = 125

→ p^3 + 1/p^3 + 3(p + 1/p) = 125

→ p^3 + 1/p^3 + 3(5) = 125

→ p^3 + 1/p^3 + 15 = 125

→ p^3 + 1/p^3 = 125 - 15

→ p^3 + 1/p^3 = 115

Hence, proved:

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