Question

solve the problems please ​

Answer 1

Solution

Given :-

• x = (3 + √5)/2

Find :-

• Value of x³ + 1/x³ .

Show That:-

• x + 1/x = 3.

Explanation

First Calculate, 1/x

==> 1/x = 1/(3+√5)/2

==> 1/x = 2/(3+√5)

Rationalize Denominator,

==> 1/x = 2(3-√5)/(3-√5)(3+√5)

==> 1/x = 2(3-√5)/(3²-√5²)

==> 1/x = 2(3-√5)/(9-5)

==> 1/x = 2(3-√5)/4

==> 1/x = (3-√5)/2.

Show, now take L.H.S.

= x + 1/x

keep value,

= (3+√5)/2 + (3-√5)/2

= [(3+√5) + (3-√5)]/2

= [(3+√5 + 3-√5)/2

= 6/2

= 3

R.H.S.

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Now, Calculate value of ( x³ + 1/x³)

==> (x³ + 1/x³) = [(3+√5)/2]³ + [(3-√5)/2]³

==> (x³ + 1/x³) = (3+√5)³/8 + (3-√5)³/8

==> (x³ + 1/x³) = [(3+√5)³+(3-√5)³]/8

==> (x³ + 1/x³) = [ (3³ + √5³ + 3×3²×√5 + 3×3×√5²)+(3³-√5³-3×3²×√5 + 3×3×√5²)]/8

==> (x³ + 1/x³) = [( 9 + 5√5 + 27√5 + 45)+(9 - 5√5 - 27√5+45)]/8

==> (x³ + 1/x³) = [ (54+32√5)+(54-32√5)]/8

==> (x³ + 1/x³) = 108)/8

==> (x³ + 1/x³) = 27/2.

Hence

• Value of (x³ + 1/x³) will be = 27/2 .

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