(x^a+b)^2(x^b+c)^2(x^c+a)^2/(x^a X^b X^c)^4

4 min read Jun 17, 2024
(x^a+b)^2(x^b+c)^2(x^c+a)^2/(x^a X^b X^c)^4

Simplifying the Expression: (x^a+b)^2(x^b+c)^2(x^c+a)^2/(x^a x^b x^c)^4

This article explores the simplification of the complex mathematical expression: (x^a+b)^2(x^b+c)^2(x^c+a)^2/(x^a x^b x^c)^4

The key to simplifying this expression lies in understanding the properties of exponents and factorization.

Step 1: Expanding the Squares

We begin by expanding the squares in the numerator. Recall that (a+b)^2 = a^2 + 2ab + b^2. Applying this to each term, we get:

  • (x^a+b)^2 = x^(2a) + 2x^a*b + b^2
  • (x^b+c)^2 = x^(2b) + 2x^b*c + c^2
  • (x^c+a)^2 = x^(2c) + 2x^c*a + a^2

Step 2: Simplifying the Denominator

The denominator can be simplified using the rule x^m * x^n = x^(m+n). Therefore,

(x^a x^b x^c)^4 = x^(4a+4b+4c)

Step 3: Combining Terms

Now, we have:

(x^(2a) + 2x^ab + b^2)(x^(2b) + 2x^bc + c^2)(x^(2c) + 2x^c*a + a^2) / x^(4a+4b+4c)

To further simplify, we need to multiply the terms in the numerator. This will result in a long expression with many terms. However, we can observe that some terms will cancel out due to the denominator.

Step 4: Cancellation and Final Expression

After multiplying the terms in the numerator and carefully observing the terms, we can cancel out many terms due to the denominator x^(4a+4b+4c). This leaves us with a simplified expression.

The final simplified expression is:

1 + 2(b/x^a + c/x^b + a/x^c) + (b^2/x^(2a) + c^2/x^(2b) + a^2/x^(2c)) + 2(bc/x^(a+b) + ac/x^(a+c) + ab/x^(b+c))


By applying the basic properties of exponents, expanding squares, and simplifying through cancellation, we have successfully simplified the complex expression. The final expression is more manageable and easier to understand, providing a clearer representation of the original equation.

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