Question

the greatest integer less than or equal to root 3+2 whole power 5

Answer 1

Given :  (√3 + 2)⁵

To Find : greatest integer less than or equal to  (√3 + 2)⁵

Solution:

( 2 + √3)⁵  + (2 - √3)⁵

= 2( ⁵C₀2⁵  + ⁵C₂2³(√3)² + ⁵C₄2(√3)⁴ )     ( all other terms cancelled because of opposite sign)

= 2( 32  +10*24 + 5*18)

= 2(32 +  240 + 90)

= 2(362)

= 724

( 2 + √3)⁵ = 724 - (2 - √3)⁵

0 < 2 - √3 < 1

0 <  (2 - √3)⁵   < 1

=>724 - 1      <  ( 2 + √3)⁵  <  724 - 0

=>  723 <  ( 2 + √3)⁵  < 724

Hence  greatest integer less than or equal to (√3 + 2)⁵  = 723

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