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1. if a - 1/a =4 ; find :

(i)a^2 + 1/a^2 (ii)a^4 + 1/a^4 (iii)a^3 - 1/a^3

2. if a - 1/a =3 ; find :

(i)a^2 + 1/a^2 (ii)a^3 - 1/a^3

1. Multiplying through by a gives a² - 4a - 1 = 0.

This means, by quadratic equation, that a = (4ħsqrt(16+4))/2.

That is the same as a = (4ħsqrt(20))/2.

Since sqrt(20) = sqrt(4*5) = 2*sqrt(5), both 4 and that term can be divided by 2.

This gives a = 2 ħ sqrt(5).

Note that this looks like it means that a² = 9 ħ 4*sqrt(5) and a³ = 38 ħ 17*sqrt(5).

Use the '+' term to find one result and the '-' term to find the other.

To simplify the solution, multiply the numerator and denominator of the fraction with the square-root in the denominator by the conjugate of the denominator. This will make the denominator an integer (no square-roots any more).

2. Multiplying through by a gives a² - 3a - 1 = 0.

The quadratic equation tells us that a = (3ħsqrt(9+4))/2 = (3ħsqrt(13))/2.

Put that in for a in (i) and (ii).

Note that it looks like a² = (22ħ6*root(13))/4 and a³ = (144ħ40*sqrt(13))/8.

Again, use the conjugate of the denominator on the fraction with the square-root in the denominator.

Algebra

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