Write the following expression into its simplest form?
(1/x-1-1/xsquare-1-1/x+1-4/xsquare+1)/
xraise to 4/xsquare-1
Let's use the symbol ^ to mean "raised to the power of" when using exponents.
I'll try to decipher your expression:
[1/(x - 1) - 1/(x^2 - 1) - 1/(x + 1) - 4/(x^2 + 1)] / [x^4/(x^2 - 1)]
Try to factor whenever you can:
[1/(x - 1) - 1/(x + 1)(x - 1) - 1/(x + 1) - 4/(x^2 + 1)] / [x^4/(x + 1)(x - 1)]
Let's find a common denominator, which is (x^2 + 1)(x + 1)(x - 1).
Using the common denominator in the numerator:
[1(x^2 + 1)(x + 1)/(x^2 + 1)(x + 1)(x - 1) - 1(x^2 + 1)/(x^2 + 1)(x + 1)(x - 1) - 1(x^2 + 1)(x - 1)/(x^2 + 1)(x + 1)(x - 1) - 4(x + 1)(x - 1)/(x^2 + 1)(x + 1)(x - 1)] / [x^4/(x + 1)(x - 1)]
Simplifying:
[(x^2 + 1)(x + 1) - (x^2 + 1) - (x^2 + 1)(x - 1) - 4(x + 1)(x - 1)]/(x^2 + 1)(x + 1)(x - 1) * (x + 1)(x - 1)/x^4 --> Invert the fraction and multiply (I'm using * to mean multiply).
[(x^2 + 1)(x + 1) - (x^2 + 1) - (x^2 + 1)(x - 1) - 4(x + 1)(x - 1)]/ x^4(x^2 + 1)
[(x^3 + x + x^2 + 2 - x^2 - 1 - x^3 - x + x^2 + 1 - 4x^2 + 4)]/[(x^6 + x^4)]
Combining like terms:
(-3x^2 + 6)/(x^6 + x^4)
And that's as far as we can go on this one if I haven't missed anything. I hope this is what you were asking.
To simplify the given expression, we follow these steps:
1. Rewrite the expression using proper notation:
(1/(x - 1) - 1/(x^2 - 1) - 1/(x + 1) - 4/(x^2 + 1)) / (x^4/(x^2 - 1))
2. Factor the denominators, if possible, for easier calculations:
[1/(x - 1) - 1/((x + 1)(x - 1)) - 1/(x + 1) - 4/(x^2 + 1)] / (x^4/((x + 1)(x - 1)))
3. Find a common denominator for all the fractions:
The common denominator is (x + 1)(x - 1)(x^2 + 1).
4. Simplify the numerator using the common denominator:
[1(x^2 + 1)(x + 1) - 1(x^2 + 1) - 1(x^2 + 1)(x - 1) - 4(x + 1)(x - 1)] / (x^4(x + 1)(x - 1))
5. Simplify the numerator by combining like terms:
((x^2 + 1)(x + 1) - (x^2 + 1) - (x^2 + 1)(x - 1) - 4(x + 1)(x - 1)) / (x^4(x + 1)(x - 1))
6. Continue simplifying the numerator:
(x^3 + x + x^2 + 2 - x^2 - 1 - x^3 - x + x^2 + 1 - 4x^2 + 4) / (x^4(x + 1)(x - 1))
7. Combine like terms again in the numerator:
(-3x^2 + 6) / (x^4(x + 1)(x - 1))
This is the simplified form of the given expression: (-3x^2 + 6) / (x^4(x + 1)(x - 1))