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

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

Simplifying the Expression: (x^a/x^b)^a^2+ab+b^2 (x^b/x^c)^b^2+bc+c^2 (x^c/x^a)^c^2+ca+a^2

This expression involves multiple exponents and fractions, making it seem complex at first glance. However, we can simplify it significantly using the rules of exponents.

Applying the Rules of Exponents

Let's break down the simplification step-by-step:

  1. Dividing exponents with the same base:
    Recall that x^m / x^n = x^(m-n). Applying this rule to each fraction within the expression:

    • (x^a / x^b) = x^(a-b)
    • (x^b / x^c) = x^(b-c)
    • (x^c / x^a) = x^(c-a)

  2. Raising a power to another power: We know that (x^m)^n = x^(m*n). Applying this to each term:

    • (x^(a-b))^(a^2 + ab + b^2) = x^((a-b)(a^2 + ab + b^2))
    • (x^(b-c))^(b^2 + bc + c^2) = x^((b-c)(b^2 + bc + c^2))
    • (x^(c-a))^(c^2 + ca + a^2) = x^((c-a)(c^2 + ca + a^2))

  3. Multiplying the exponents: Let's expand the exponents in each term:

    • (a-b)(a^2 + ab + b^2) = a^3 - b^3
    • (b-c)(b^2 + bc + c^2) = b^3 - c^3
    • (c-a)(c^2 + ca + a^2) = c^3 - a^3

  4. Combining the terms: Now, the expression becomes: x^(a^3 - b^3) * x^(b^3 - c^3) * x^(c^3 - a^3)

    Applying the rule x^m * x^n = x^(m+n), we get:

    x^(a^3 - b^3 + b^3 - c^3 + c^3 - a^3)

  5. Simplifying further: Notice that all the exponents cancel out, leaving us with:

    x^0

Final Result

The simplified form of the expression (x^a/x^b)^a^2+ab+b^2 (x^b/x^c)^b^2+bc+c^2 (x^c/x^a)^c^2+ca+a^2 is 1.

This simplification demonstrates the power of understanding the rules of exponents and using them strategically to simplify complex expressions.

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