Question

If a = 3+2√2 then find the value of a^3+1/a^3​

Answer 1

GIVEn

If a = 3+2√2 then find the value of a^3+1/a^3

SOLUTIOn

• a = 3 + 2√2

→ 1/a = 3 + 2√2

Rationalize its denominator

→ 1/a = 1/3 + 2√2 × 3 - 2√2/3 - 2√2

→ 1/a = 3 - 2√2 /(3)² - (2√2)²

→ 1/a = 3 - 2√2/9 - 8

→ 1/a = 3 - 2√2

Now, value of a + 1/a

→ a + 1/a

→ 3 + 2√2 + 3 - 2√2

→ 6

So, (a + 1/a) = 6

_____________________

→ (a + 1/a) = 6

cube both side

→ (a + 1/a)³ = 6³

→ a³ + 1/a³ + 3*a*1/a(a + 1/a) = 216

→ a³ + 1/a³ + 3*6 = 216

→ a³ + 1/a³ + 18 = 216

→ a³ + 1/a³ = 216 - 18

→ a³ + 1/a³ = 198

Therefore, the value of a³ + 1/a³ is 198

_____________________

Answer 2

${\purple{\underline{\underline{\large{\mathtt{Answer:}}}}}}$

Given:

• We have been given that a = 3+2√2.

To Find:

• We need to find the value of a³+1/a³.

Solution:

As it is given that a = 3+2√2, so let us try to find the value of a + 1/a.

=> 3 + 2√2 + 1/3+2√2

=> 3 + 2√2 + ( 3 - 2√2 ) / ( 9 - 8 )

=> 3 + 2√2 + 3 - 2√2

=> 3 + 3

=> 6_____(1)

Now, we need to find the value of a³+1/a³.

We have, 6³= (a + 1/a)³ [From equation 1]

=> a³ + 3 a² (1/a) + 3 a (1/a)² + 1/a³ = 216

=> a³ + 3a + 3/a + 1/a³

=> a³ + 1/a³ + 3 ( a + 1/a )

=> a³ + 1/a³ + 3 (6)

=> a³ + 1/a³ + 18

=> a³ + 1/a³ = 216 - 18

=> a³ + 1/a³ = 198

Therefore, the value of a³ + 1/a³ is 198.