Certainly! Let's simplify the expression step-by-step:
[tex]\[
\left(\frac{x^{a+b}}{x^c}\right)^{a-b}\left(\frac{x^{b+c}}{x^a}\right)^{b-c}\left(\frac{x^{c+a}}{x^b}\right)^{c-a}
\][/tex]
First, let's simplify the base of each term inside the parentheses:
### Simplifying Each Base Term
#### Term 1:
[tex]\[
\left(\frac{x^{a+b}}{x^c}\right)^{a-b}
\][/tex]
Using the property of exponents [tex]\(\frac{x^m}{x^n} = x^{m-n}\)[/tex]:
[tex]\[
\left(x^{(a+b) - c}\right)^{a-b} = \left(x^{a+b-c}\right)^{a-b}
\][/tex]
#### Term 2:
[tex]\[
\left(\frac{x^{b+c}}{x^a}\right)^{b-c}
\][/tex]
Using the same exponent property:
[tex]\[
\left(x^{(b+c) - a}\right)^{b-c} = \left(x^{b+c-a}\right)^{b-c}
\][/tex]
#### Term 3:
[tex]\[
\left(\frac{x^{c+a}}{x^b}\right)^{c-a}
\][/tex]
Using the same exponent property:
[tex]\[
\left(x^{(c+a) - b}\right)^{c-a} = \left(x^{c+a-b}\right)^{c-a}
\][/tex]
### Combining the Simplified Terms
Now we have:
[tex]\[
\left(x^{a+b-c}\right)^{a-b} \left(x^{b+c-a}\right)^{b-c} \left(x^{c+a-b}\right)^{c-a}
\][/tex]
Applying the power of a power property [tex]\((x^m)^n = x^{mn}\)[/tex]:
#### Term 1:
[tex]\[
\left(x^{a+b-c}\right)^{a-b} = x^{(a+b-c)(a-b)}
\][/tex]
#### Term 2:
[tex]\[
\left(x^{b+c-a}\right)^{b-c} = x^{(b+c-a)(b-c)}
\][/tex]
#### Term 3:
[tex]\[
\left(x^{c+a-b}\right)^{c-a} = x^{(c+a-b)(c-a)}
\][/tex]
So, the expression now is:
[tex]\[
x^{(a+b-c)(a-b)} \cdot x^{(b+c-a)(b-c)} \cdot x^{(c+a-b)(c-a)}
\][/tex]
### Combining the Exponents
Using the property of exponents [tex]\(x^m \cdot x^n \cdot x^p = x^{m+n+p}\)[/tex]:
[tex]\[
x^{(a+b-c)(a-b) + (b+c-a)(b-c) + (c+a-b)(c-a)}
\][/tex]
The simplified expression inside the exponent:
[tex]\[
(-a + b + c)(b - c) + (a - b + c)(-a + c) + (a + b - c)(a - b)
\][/tex]
Hence, the simplified form of the given expression is:
[tex]\[
(x^{(-a + b + c)})^{(b - c)} \cdot (x^{(a - b + c)})^{(-a + c)} \cdot (x^{(a + b - c)})^{(a - b)}
\][/tex]
