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# Algebra/expansion ( including substitution )

Question
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

Volunteer

#### Scott A Wilson

##### Expertise

Any algebraic question you've got. That includes question that are linear, quadratic, exponential, etc.

##### Experience

I have solved story problems, linear equations, parabolic equations. I have also solved some 3rd order equations and equations with multiple variables.

Publications
Documents at Boeing in assistance on the manufacturiing floor.

Education/Credentials
MS at math OSU in mathematics at OSU, 1986. BS at OSU in mathematical sciences (math, statistics, computer science), 1984.

Awards and Honors
Both my BS and MS degrees were given with honors.

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Students in a wide variety of areas since the 80's; over 1,000 of them have been in algebra.