Answer :

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.