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(x^a/x^b)^1/ab*(x^b/x^c)^1/bc*(x^c/x^a)^1/ca

4 min read Jun 17, 2024
(x^a/x^b)^1/ab*(x^b/x^c)^1/bc*(x^c/x^a)^1/ca

Simplifying the Expression: (x^a/x^b)^1/ab*(x^b/x^c)^1/bc*(x^c/x^a)^1/ca

This article explores the simplification of the expression (x^a/x^b)^1/ab(x^b/x^c)^1/bc(x^c/x^a)^1/ca**. We will use the properties of exponents to break down the expression and arrive at a simplified form.

Using Properties of Exponents

The key to simplifying this expression lies in using the following properties of exponents:

  • (x^m)^n = x^(m*n)
  • x^m/x^n = x^(m-n)

Let's apply these properties step by step.

  1. Simplify each term individually:

    • (x^a/x^b)^1/ab = (x^(a-b))^1/ab = x^((a-b)/ab)
    • (x^b/x^c)^1/bc = (x^(b-c))^1/bc = x^((b-c)/bc)
    • (x^c/x^a)^1/ca = (x^(c-a))^1/ca = x^((c-a)/ca)

  2. Combine the simplified terms:

    • The entire expression now becomes: x^((a-b)/ab) * x^((b-c)/bc) * x^((c-a)/ca)

  3. Apply the property: x^m * x^n = x^(m+n):

    • This gives us: x^((a-b)/ab + (b-c)/bc + (c-a)/ca)

  4. Find a common denominator for the exponents:

    • The common denominator for the fractions is abc. This gives us: x^((ac(a-b) + ab(b-c) + bc(c-a))/abc)

  5. Simplify the numerator:

    • Expanding the numerator, we get: x^((a^2c - abc + ab^2 - abc + bc^2 - abc)/abc)
    • Combining like terms: x^((a^2c + ab^2 + bc^2 - 3abc)/abc)

  6. Factor out a 'c' from the numerator:

    • This results in: x^(c(a^2 + b^2 + c^2 - 3ab)/abc)

  7. Cancel out the 'c' in the numerator and denominator:

    • This leaves us with: x^((a^2 + b^2 + c^2 - 3ab)/ab)

Conclusion

Therefore, the simplified form of the expression (x^a/x^b)^1/ab(x^b/x^c)^1/bc(x^c/x^a)^1/ca** is x^((a^2 + b^2 + c^2 - 3ab)/ab). This expression can be further analyzed or evaluated depending on the values of a, b, and c.

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