Simplify:
[(x^-3)-x]/[(x^-2)-1]
x^3-x = x(x^2-1)
I don't get it...
sure you do. factor out x^2-1 top and bottom
but don't you have to get rid of the negative exponents somehow..?
sorry. I missed the negative exponents.
(x^-3 - x)/(x^(-2)-1)
(1 - x^4)/x^3 / (1-x^2)/x^2
(1-x^4)/x^3 * x^2/(1-x^2)
(1-x^2)(1+x^2) / x(1-x^2)
(1+x^2)/x
How did you go from the original question to the 1st step?
1/x^3 - x
over a common denominator of x^3, x = x^4/x^3
ohmygosh.. i'm an idiot.. thank you! you're the bestt!(:
To simplify the expression [(x^-3)-x]/[(x^-2)-1], we can follow these steps:
Step 1: Simplify the numerator.
The numerator is (x^-3) - x. To simplify this, we need to find a common denominator for x^-3 and x. The common denominator can be expressed as x^-3 since x^-3 already contains x. Therefore, the numerator becomes (1 - x^(4-3)) = (1 - x^1) = (1 - x).
Step 2: Simplify the denominator.
The denominator is (x^-2) - 1. In this case, we don't need to find a common denominator since there are no common factors between x^-2 and 1 that can simplify the expression.
Step 3: Simplify the fraction.
Now that we have simplified the numerator and denominator separately, we can simplify the whole expression. The expression becomes (1 - x)/[(x^-2) - 1].
Step 4: Rationalize the denominator.
To rationalize the denominator, we need to eliminate the negative exponent in x^-2. We can do this by multiplying both the numerator and denominator by x^2.
(1 - x)/[(x^-2) - 1] * (x^2)/(x^2)
Multiplying the numerator and denominator gives us:
(x^2 - x^3)/(1 - x^2)
This is the simplified form of the expression [(x^-3)-x]/[(x^-2)-1].