To evaluate the expression [tex]\(\frac{\left(x^{a+b}\right)^2 \cdot \left(x^{b+c}\right)^2 \cdot \left(x^{c+a}\right)^2}{\left(x^a \cdot x^b \cdot x^c\right)^4}\)[/tex], we will simplify both the numerator and the denominator separately and then combine the results.
First, let's simplify the numerator:
[tex]\[
\left(x^{a+b}\right)^2 \cdot \left(x^{b+c}\right)^2 \cdot \left(x^{c+a}\right)^2
\][/tex]
This can be rewritten by using the power rule [tex]\((x^m)^n = x^{m \cdot n}\)[/tex]:
[tex]\[
x^{2(a+b)} \cdot x^{2(b+c)} \cdot x^{2(c+a)}
\][/tex]
Next, we add the exponents together (since [tex]\(x^m \cdot x^n = x^{m+n}\)[/tex]):
[tex]\[
x^{2(a+b) + 2(b+c) + 2(c+a)}
\][/tex]
Now expand and combine like terms inside the exponent:
[tex]\[
2(a+b) + 2(b+c) + 2(c+a) = 2a + 2b + 2b + 2c + 2c + 2a = 4a + 4b + 4c
\][/tex]
Therefore, the numerator simplifies to:
[tex]\[
x^{4a + 4b + 4c}
\][/tex]
Next, let's simplify the denominator:
[tex]\[
\left(x^a \cdot x^b \cdot x^c\right)^4
\][/tex]
This can be rewritten by using the multiplication rule [tex]\((x^m \cdot x^n = x^{m+n})\)[/tex]:
[tex]\[
x^{a+b+c}
\][/tex]
Raise this expression to the fourth power using the power rule [tex]\((x^m)^n = x^{m \cdot n}\)[/tex]:
[tex]\[
(x^{a+b+c})^4 = x^{4(a+b+c)}
\][/tex]
Now, put the simplified numerator and denominator together:
[tex]\[
\frac{x^{4a + 4b + 4c}}{x^{4(a+b+c)}}
\][/tex]
Notice that [tex]\(4(a+b+c) = 4a + 4b + 4c\)[/tex], so the exponent in the numerator and denominator are the same. We can then use the rule for dividing powers with the same base: [tex]\(\frac{x^m}{x^n} = x^{m-n}\)[/tex]:
[tex]\[
x^{4a + 4b + 4c - 4(a+b+c)} = x^0
\][/tex]
Since any non-zero number raised to the power of 0 is 1:
[tex]\[
x^0 = 1
\][/tex]
Therefore, the value of the given expression is:
[tex]\[
1
\][/tex]
