How do you simplify # [ (a^2 - b^2)^3 + (b^2 - c^2)^3 + (c^2 - a^2)^3 ] / [ (a^4 - b^4)^3 + (b^4 - c^4)^3 + (c^4 - a^4)^3]#?

Answer 1

#((a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3)/((a^4-b^4)^3+(b^4-c^4)^3+(c^4-a^4)^3)=1/((a^2+b^2)(b^2+c^2)(c^2+a^2))#

excluding any of #a=+-b#, #b=+-c#, #c=+-a#.

Notice that:

#(A-B)^3+(B-C)^3+(C-A)^3#
#=(color(red)(cancel(color(black)(A^3)))-3A^2B+3AB^2-color(red)(cancel(color(black)(B^3))))+(color(red)(cancel(color(black)(B^3)))-3B^2C+3BC^2-color(red)(cancel(color(black)(C^3))))+(color(red)(cancel(color(black)(C^3)))-3C^2A+3CA^2-color(red)(cancel(color(black)(A^3))))#
#=3(AB^2-A^2B+BC^2-B^2C+CA^2-C^2A)#
#=3(A-B)(B-C)(C-A)#

Note also:

#a^4-b^4 = (a^2-b^2)(a^2+b^2)#, etc.

So:

#((a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3)/((a^4-b^4)^3+(b^4-c^4)^3+(c^4-a^4)^3)#
#=(color(red)(cancel(color(black)(3)))(a^2-b^2)(b^2-c^2)(c^2-a^2))/(color(red)(cancel(color(black)(3)))(a^4-b^4)(b^4-c^4)(c^4-a^4))#
#=(color(red)(cancel(color(black)((a^2-b^2))))color(red)(cancel(color(black)((b^2-c^2))))color(red)(cancel(color(black)((c^2-a^2)))))/(color(red)(cancel(color(black)((a^2-b^2))))(a^2+b^2)color(red)(cancel(color(black)((b^2-c^2))))(b^2+c^2)color(red)(cancel(color(black)((c^2-a^2))))(c^2+a^2))#
#=1/((a^2+b^2)(b^2+c^2)(c^2+a^2))#
excluding any of #a=+-b#, #b=+-c#, #c=+-a#
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Answer 2

To simplify the expression, we can factorize the numerator and denominator using the identity a^3 - b^3 = (a - b)(a^2 + ab + b^2). After factorizing, we can cancel out common factors. The simplified expression is:

[(a^2 - b^2)^3 + (b^2 - c^2)^3 + (c^2 - a^2)^3] / [(a^4 - b^4)^3 + (b^4 - c^4)^3 + (c^4 - a^4)^3]

= [(a - b)^3(a^2 + ab + b^2) + (b - c)^3(b^2 + bc + c^2) + (c - a)^3(c^2 + ac + a^2)] / [(a^2 - b^2)^3(a^2 + b^2) + (b^2 - c^2)^3(b^2 + c^2) + (c^2 - a^2)^3(c^2 + a^2)]

= [(a - b)^3(a^2 + ab + b^2) + (b - c)^3(b^2 + bc + c^2) + (c - a)^3(c^2 + ac + a^2)] / [(a - b)^3(a + b)(a^2 + b^2) + (b - c)^3(b + c)(b^2 + c^2) + (c - a)^3(c + a)(c^2 + a^2)]

= [(a - b)(a^2 + ab + b^2) + (b - c)(b^2 + bc + c^2) + (c - a)(c^2 + ac + a^2)] / [(a - b)(a + b)(a^2 + b^2) + (b - c)(b + c)(b^2 + c^2) + (c - a)(c + a)(c^2 + a^2)]

= (a^2 + ab + b^2 + b^2 + bc + c^2 + c^2 + ac + a^2) / (a^3 - b^3 + a^2b - ab^2 + b^3 - c^3 + b^2c - bc^2 + c^3 - a^3 + c^2a - ac^2)

= (2a^2 + 2b^2 + 2c^2 + ab + ac + bc) / (2a^2b - 2ab^2 + 2b^2c - 2bc^2 + 2c^2a - 2ac^2)

= (a^2 + b^2 + c^2 + ab + ac + bc) / (a^2b - ab^2 + b^2c - bc^2 + c^2a - ac^2)

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Answer 3

To simplify the expression [\frac{{(a^2 - b^2)^3 + (b^2 - c^2)^3 + (c^2 - a^2)^3}}{{(a^4 - b^4)^3 + (b^4 - c^4)^3 + (c^4 - a^4)^3}},] we can use the identities (a^3 - b^3 = (a - b)(a^2 + ab + b^2)) and (a^3 + b^3 = (a + b)(a^2 - ab + b^2)) to factor the numerator and denominator.

Factoring the terms in the numerator, we get: [(a^2 - b^2)^3 = (a^2 - b^2)(a^4 - 2a^2b^2 + b^4),] [(b^2 - c^2)^3 = (b^2 - c^2)(b^4 - 2b^2c^2 + c^4),] [(c^2 - a^2)^3 = (c^2 - a^2)(c^4 - 2c^2a^2 + a^4).]

Similarly, factoring the terms in the denominator, we get: [(a^4 - b^4)^3 = (a^4 - b^4)(a^8 - 3a^4b^4 + b^8),] [(b^4 - c^4)^3 = (b^4 - c^4)(b^8 - 3b^4c^4 + c^8),] [(c^4 - a^4)^3 = (c^4 - a^4)(c^8 - 3c^4a^4 + a^8).]

Now, substituting these factorizations back into the expression, we can cancel out the common factors: [\frac{{(a^2 - b^2)(a^4 - 2a^2b^2 + b^4) + (b^2 - c^2)(b^4 - 2b^2c^2 + c^4) + (c^2 - a^2)(c^4 - 2c^2a^2 + a^4)}}{{(a^4 - b^4)(a^8 - 3a^4b^4 + b^8) + (b^4 - c^4)(b^8 - 3b^4c^4 + c^8) + (c^4 - a^4)(c^8 - 3c^4a^4 + a^8)}}.]

After expanding and simplifying the terms, we can see if any common factors can be canceled further. However, the expression at this point doesn't simplify further without additional information or specific values assigned to (a), (b), and (c).

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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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