Question

If a is not equal to 0 and a-1/a=3; find : (1) a^2 +1/a^2 (2) a^3-1/a^3

Answer 1

Answer:

11 and 36

Step-by-step explanation:

      ⇒ a - 1 / a = 3

 Square on both sides:

⇒ ( a - 1 / a )^2 = 3^2

⇒ a^2 + 1 / a^2 - 2( a * 1 / a ) = 9

⇒ a^2 + 1 / a^2 - 2( 1 ) = 9

⇒ a^2 + 1 / a^2 - 2 = 9

⇒ a^2 + 1 / a^2 = 9 + 2 = 11

         Cube on both sides of a - 1 / a:

⇒ ( a - 1 / a )^3 = 3^3

⇒ a^3 - 1 / a^3 - 3( a * 1 / a )( a - 1 / a ) = 27

⇒ a^3 - 1 / a^3 - 3( 1 )( a - 1 / a ) = 27

⇒ a^3 - 1 / a^3 - 3( 3 ) = 27

⇒ a^3 - 1 / a^3 - 9 = 27

⇒ a^3 - 1 / a^3 = 27 + 9 = 36

Answer 2

$↠\red{a - \frac{1}{a} = 3} \\↠ \red{( a - \frac{1}{a}) {}^{2} = {3}^{2} } \\↠ {a}^{2} + \frac{1}{ {a}^{2} } - 2 \times a \times \frac{1}{a} = 9 \\↠ {a}^{2} + \frac{1}{ {a}^{2} } - 2 = 9 \\ ↠ {a}^{2} + \frac{1}{ {a}^{2} } = 9 + 2 = 11$

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$↠\red{a - \frac{1}{a} = 3} \\↠ \red{(a - \frac{1}{a})^{3} = {3}^{3}} \\↠ {a}^{3} - \frac{1}{{a}^{3}} - 3\times a\times \frac{1}{a} (a- \frac{1}{a} )= 27 \\↠ {a}^{3} - \frac{1}{{a}^{3}} -3(3)=27 \\ ↠{a}^{3} - \frac{1}{{a}^{3}} -9=27 \\ ↠{a}^{3} - \frac{1}{{a}^{3}} =27+9=36$

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∴ $\red{{a}^{2} + \frac{1}{ {a}^{2} } = 9 + 2 = 11}$

∴ $\red{{a}^{3} - \frac{1}{{a}^{3}} =27+9=36}$