How do you add or subtract #(y^2-5)/(y^4-81) + 4/(81-y^4)#?

Answer 1

We can rewrite this sum as

#(y^2-5)/(y^4-81)+4/((-1)(y^4-81))#
We can find the lowest common denominator, which, then, is #(-1)(y^4-81)#

Using the l.c.d.:

#((-1)(y^2-5)+4)/((-1)(y^4-81))#=#=(-y^2+9)/(-y^4+81)#

That can be your final answer, but let's draw attention to the fact we have factorable functions there.

Let's find the roots of #-y^2+9# by equaling this to zero and, then, factoring it:
#9=y^2# #y=+-3#, which means #y-3=0# and #y+3=0#. The two roots have been found.

Now, as for the denominator:

#y^4=81#

Let's just go slowly here and take the square root of both sides.

#sqrt(y^4)=sqrt(81)#
#y^2=9#
#y=sqrt(9)=+-3#
So, here, as we are dealing with a polinomial of 4#th# degree, we can state that #-y^4+81=(x+3)(x-3)(x+3)(x-3)#, because we have #y^color(green)4# and not only #y^color(red)2# as in the previous factor.

Therefore, your "final" answer can be rewritten as

#(cancel(x+3)cancel(x-3))/(cancel(x+3)cancel(x-3)(x+3)(x-3))#
Finalfinal answer, then: #1/((x+3)(x-3))#
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Answer 2

To add or subtract the given expressions, we need to find a common denominator. In this case, the common denominator is (y^4 - 81).

To add the fractions, we multiply the numerator and denominator of the first fraction by (81 - y^4), and the numerator and denominator of the second fraction by (y^4 - 81).

This gives us:

[(y^2 - 5)(81 - y^4)] / [(y^4 - 81)(81 - y^4)] + [4(y^4 - 81)] / [(y^4 - 81)(81 - y^4)]

Simplifying the numerators, we have:

[(81y^2 - 405 - 81y^4 + 5y^2)] / [(y^4 - 81)(81 - y^4)] + [4y^4 - 324] / [(y^4 - 81)(81 - y^4)]

Combining like terms in the numerators, we get:

[(86y^2 - 81y^4 - 405)] / [(y^4 - 81)(81 - y^4)] + [4y^4 - 324] / [(y^4 - 81)(81 - y^4)]

Now, we can combine the fractions by adding the numerators:

[(86y^2 - 81y^4 - 405 + 4y^4 - 324)] / [(y^4 - 81)(81 - y^4)]

Simplifying the numerator further:

[(86y^2 - 77y^4 - 729)] / [(y^4 - 81)(81 - y^4)]

Therefore, the simplified expression is:

(86y^2 - 77y^4 - 729) / (y^4 - 81)(81 - y^4)

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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