if, then problem help! if (a+1/a)^2 = 3, then a^3 +1/a^3 equals what?

[jacknowlin]

The question asks:

if (a+1/a)^2 = 3, then a^3 +1/a^3 equals what?

I started by trying to simplify the if equation so that I could find what a equals in order to plug it into the second equation, but I feel like im doing the algebra wrong:

I've done:

square root(a +1/a) = square root(3)

but I dont know if you are even able to simplify the square root (a+ 1/a) and dont even know if im taking the correct steps

 

[Deleted member 4993]

The question asks:

if (a+1/a)^2 = 3, then a^3 +1/a^3 equals what?

I started by trying to simplify the if equation so that I could find what a equals in order to plug it into the second equation, but I feel like im doing the algebra wrong:

I've done:

square root(a +1/a) = square root(3)

but I dont know if you are even able to simplify the square root (a+ 1/a) and dont even know if im taking the correct steps

Hint:

x^3 + y^3 = (x + y)(x^2 -xy + y^2)

 

[JeffM]

The question asks:

if (a+1/a)^2 = 3, then a^3 +1/a^3 equals what?

I started by trying to simplify the if equation so that I could find what a equals in order to plug it into the second equation, but I feel like im doing the algebra wrong:

I've done:

square root(a +1/a) = square root(3)

but I dont know if you are even able to simplify the square root (a+ 1/a) and dont even know if im taking the correct steps

OK, your first idea was absolutely on target: solve for a and then substitute the value or values found into the second expression. Perfect.

There was nothing inherently wrong in the square root idea, but it is a dead end as you found. One of the things that is not stressed enough in teaching math is to try different things. There are frequently a number of techniques that may be applicable to a problem. You can't give up when one turns out to be not the one for a particular problem. You have more than one tool to work with. Now, taking the square root of an expression is seldom a useful tool unless the the square root symbol simplifies out. That is not the case here. So this tool is not useful for this problem.

Do you have another tool in your tool box that relates directly to squares? You do. It is a frequently useful tool called the quadratic formula. But to use it, you need to expand the expression to have actual squares.

\(\displaystyle \left ( \dfrac{a + 1}{a} \right )^2 = 3 \implies \dfrac{(a + 1)^2}{a^2} = 3 \implies\)

\(\displaystyle \dfrac{a^2 + 2a + 1}{a^2} = 3 \implies WHAT?\)

EDIT: SK has given you a hint for a different approach that does involve the square root. It is very clever, but I do not think it is as intuitively obvious. Nevertheless, it is a lot easier than my approach.

 

[pka]

The question asks: if (a+1/a)^2 = 3, then a^3 +1/a^3 equals what?

Hint: x^3 + y^3 = (x + y)(x^2 -xy + y^2)

Did you consider Mr. Khan's hint to heart? \(\displaystyle {a^3} + \dfrac{1}{{{a^3}}} = \left( {a + \dfrac{1}{a}} \right)\left( {{a^2} - 1 + \dfrac{1}{{{a^2}}}} \right)\)

Now \(\displaystyle \begin{array}{l} {a^2} + 2 + \dfrac{1}{{{a^2}}} &= {\left( {a + \dfrac{1}{a}} \right)^2}\\ &= 3\\ {a^2} + \dfrac{1}{{{a^2}}} &= ? \end{array}\)

Now you show us some work.