Sure! Let's break down the problem and factorize the given expressions step by step.
### Step-by-Step Solution:
#### 1. Factorizing the First Expression
The first expression is:
[tex]\[ 6a^3 \cdot 6^2 \cdot c - 3a^2 + 46c^2 \cdot y \][/tex]
Let's simplify and rewrite it:
[tex]\[ = 6a^3 \cdot 36 \cdot c - 3a^2 + 46c^2 \cdot y \][/tex]
[tex]\[ = 216a^3c - 3a^2 + 46c^2y \][/tex]
Now let's factorize this:
[tex]\[ 216a^3c - 3a^2 + 46c^2y \][/tex]
It's tricky to spot common factors here, so let's try polynomial factorization methods to simplify this:
Finally, the factorized form of the expression is:
[tex]\[ 216a^3c - 3a^2 + 46c^2y \][/tex]
This is already in its simplified form, so no further common factorization is possible directly here.
#### 2. Factorizing the Second Expression
The second expression is:
[tex]\[ 5^4 \cdot 6^5 + 35m^2 \cdot f^2 - 30m^5 \cdot n^4 \][/tex]
Let's simplify and rewrite this expression:
[tex]\[ 5^4 \cdot 6^5 + 35m^2 \cdot f^2 - 30m^5 \cdot n^4 \][/tex]
Calculating the constants:
[tex]\[ 5^4 = 625 \][/tex]
[tex]\[ 6^5 = 7776 \][/tex]
Rewriting the expression with constants calculated:
[tex]\[ 625 \cdot 7776 + 35m^2 f^2 - 30m^5 n^4 \][/tex]
Now let's factorize this expression:
[tex]\[ 625 \cdot 7776 = 972000 \][/tex]
Rewriting:
[tex]\[ 972000 + 35m^2 f^2 - 30m^5 n^4 \][/tex]
Group the terms in a way that can make factorization easier:
[tex]\[ 5 \cdot (7f^2 m^2 - 6m^5 n^4 + 194400) \][/tex]
Factoring out the positive constant term:
[tex]\[ 5(7m^2f^2 - 6m^5n^4 + 194400) \][/tex]
So, the final factorized form is:
[tex]\[ 5(7m^2f^2 - 6m^5n^4 + 194400) \][/tex]
Therefore, our fully simplified and factorized expressions are:
#### Final Results:
1. [tex]\( 216a^3c - 3a^2 + 46c^2y \)[/tex]
2. [tex]\( 5(7m^2f^2 - 6m^5n^4 + 194400) \)[/tex]
These results match the simplified forms shown in the problem.
