Question

ifx^2+1/x^2=7 find the value of x^3+1/x^3 by taking only positive value of x+1/x​

Answer 1

Answer:

Step-by-step explanation:

Answer 2

Answer:

18

Step-by-step explanation:

Given ,

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

Adding 2 on both sides of the equation we have :

${x}^{2} + \frac{1}{ {x}^{2} } + 2 = 7 + 2 \\ \\ \implies {x}^{2} + \frac{1}{ {x}^{2} } + 2 \times x \times \frac{1}{x} = 9 \\ \\ \implies {x}^{2} + { (\frac{1}{ {x}} })^{2} + 2 \times x \times \frac{1}{x} = 9 \\ \\ \implies( { x+ \frac{1}{x} })^{2} = 9 \\ \\ \implies \: x + \frac{1}{x} = \sqrt{9}$

→ x + 1/x = ±3

As instructed in the question , only positive value of x + 1/x is taken to find x³ + 1/x³

Therefore ,

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

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

★Identities used in this Problem are :

• a² + b² + 2ab = (a + b)²

• a³ + b³ + 3ab(a +b) = (a +b)³

★More similar formulas :

• a² + b² -2ab = (a - b)²

• a³ - b³ - 3ab(a - b) = (a - b)³