To solve the expression [tex]\(\frac{a^{20} \cdot b^{16} \cdot c^{12} \cdot a^5 \cdot b^9 \cdot c^{13}}{(a b c)^{25}}\)[/tex], let's proceed step by step:
1. Combine the powers of each variable in the numerator:
- For [tex]\(a\)[/tex]:
[tex]\[
a^{20} \cdot a^5 = a^{20 + 5} = a^{25}
\][/tex]
- For [tex]\(b\)[/tex]:
[tex]\[
b^{16} \cdot b^9 = b^{16 + 9} = b^{25}
\][/tex]
- For [tex]\(c\)[/tex]:
[tex]\[
c^{12} \cdot c^{13} = c^{12 + 13} = c^{25}
\][/tex]
Thus, the numerator simplifies to:
[tex]\[
a^{25} \cdot b^{25} \cdot c^{25}
\][/tex]
2. Simplify the denominator:
Given the expression in the denominator [tex]\((a b c)^{25}\)[/tex]:
[tex]\[
(a b c)^{25} = a^{25} \cdot b^{25} \cdot c^{25}
\][/tex]
3. Combine the numerator and denominator:
Substituting our simplified expressions:
[tex]\[
\frac{a^{25} \cdot b^{25} \cdot c^{25}}{a^{25} \cdot b^{25} \cdot c^{25}}
\][/tex]
4. Cancel out the common terms:
Since the numerator and denominator are identical, we can cancel out [tex]\(a^{25} \cdot b^{25} \cdot c^{25}\)[/tex] in both:
[tex]\[
\frac{a^{25} \cdot b^{25} \cdot c^{25}}{a^{25} \cdot b^{25} \cdot c^{25}} = 1
\][/tex]
5. Express powers explicitly:
We also observe that the result of canceling out each power would be:
- For [tex]\(a\)[/tex]:
[tex]\[
a^{25 - 25} = a^0
\][/tex]
- For [tex]\(b\)[/tex]:
[tex]\[
b^{25 - 25} = b^0
\][/tex]
- For [tex]\(c\)[/tex]:
[tex]\[
c^{25 - 25} = c^0
\][/tex]
Since any number to the power of 0 is 1:
[tex]\[
a^0 \cdot b^0 \cdot c^0 = 1 \cdot 1 \cdot 1 = 1
\][/tex]
So, the step-by-step simplification and result show that the given expression simplifies to 1.
