Answer :
### Step-by-Step Solutions:
1. Simplifying the Expression:
[tex]\[ 20 a m + 28 a b + 12 a x + 8 a h + 25 f m + 35 b f + 15 f x + 10 f h \][/tex]
This expression consists of several terms, with variables [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(f\)[/tex], [tex]\(h\)[/tex], [tex]\(m\)[/tex], and [tex]\(x\)[/tex]. Let's group terms that share common variables:
- Terms with [tex]\( a \)[/tex]: [tex]\( 20 a m + 28 a b + 12 a x + 8 a h \)[/tex]
- Terms with [tex]\( f \)[/tex]: [tex]\( 25 f m + 35 b f + 15 f x + 10 f h \)[/tex]
Combined, the simplified expression is:
[tex]\[ 20 a m + 25 f m + 28 a b + 35 b f + 8 a h + 10 f h + 12 a x + 15 f x \][/tex]
So, the simplified answer is:
[tex]\[ 20 a m + 25 f m + 28 a b + 35 b f + 8 a h + 10 f h + 12 a x + 15 f x \][/tex]
2. Combining Like Terms in the Polynomial:
[tex]\[ 8 x^3 y^3 + 48 y^4 x^5 + 96 x^7 y^5 + 64 x^9 y^6 \][/tex]
Here, there are no similar terms to combine. Therefore, the expression remains as:
[tex]\[ 8 x^3 y^3 + 48 y^4 x^5 + 96 x^7 y^5 + 64 x^9 y^6 \][/tex]
3. Simplifying the Fraction and Expression:
[tex]\[ \frac{4}{64} b^{10} y^8 - 0.81 b^4 c^6 \][/tex]
First, simplify the fractional coefficient:
[tex]\[ \frac{4}{64} = \frac{1}{16} \][/tex]
So, the expression becomes:
[tex]\[ \frac{1}{16} b^{10} y^8 - 0.81 b^4 c^6 \][/tex]
4. Simplifying the Cubic Expression:
[tex]\[ c^3 - 3^3 \][/tex]
Recall that:
[tex]\[ 3^3 = 27 \][/tex]
Thus, the simplified expression is:
[tex]\[ c^3 - 27 \][/tex]
5. Factoring the Polynomial Expression:
[tex]\[ -27 a^{12} b^6 - 9 a^{12} b^4 - a^{12} b^3 - 27 a^{12} b^5 \][/tex]
Factor out the common factor [tex]\(-a^{12}\)[/tex]:
[tex]\[ -a^{12}(27 b^6 + 9 b^4 + b^3 + 27 b^5) \][/tex]
Rearrange the terms inside the parentheses:
[tex]\[ -a^{12}(27 b^6 + 27 b^5 + 9 b^4 + b^3) \][/tex]
Factor out 27 from the first two terms:
[tex]\[ -a^{12} \times 27(b^6 + b^5) - a^{12}(9 b^4 + b^3) \][/tex]
Combine all the terms under the common factor [tex]\(-64 a^{12}\)[/tex]:
[tex]\[ -64 a^{12}(b^6 + b^5 + b^4 + b^3) \][/tex]
So, the simplified answer is:
[tex]\[ -64 a^{12}(b^6 + b^5 + b^4 + b^3) \][/tex]
### Summary:
1. [tex]\( 20 a m + 25 f m + 28 a b + 35 b f + 8 a h + 10 f h + 12 a x + 15 f x \)[/tex]
2. [tex]\( 8 x^3 y^3 + 48 y^4 x^5 + 96 x^7 y^5 + 64 x^9 y^6 \)[/tex]
3. [tex]\( \frac{1}{16} b^{10} y^8 - 0.81 b^4 c^6 \)[/tex]
4. [tex]\( c^3 - 27 \)[/tex]
5. [tex]\( -64 a^{12}(b^6 + b^5 + b^4 + b^3) \)[/tex]
Buena Suerte!!
1. Simplifying the Expression:
[tex]\[ 20 a m + 28 a b + 12 a x + 8 a h + 25 f m + 35 b f + 15 f x + 10 f h \][/tex]
This expression consists of several terms, with variables [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(f\)[/tex], [tex]\(h\)[/tex], [tex]\(m\)[/tex], and [tex]\(x\)[/tex]. Let's group terms that share common variables:
- Terms with [tex]\( a \)[/tex]: [tex]\( 20 a m + 28 a b + 12 a x + 8 a h \)[/tex]
- Terms with [tex]\( f \)[/tex]: [tex]\( 25 f m + 35 b f + 15 f x + 10 f h \)[/tex]
Combined, the simplified expression is:
[tex]\[ 20 a m + 25 f m + 28 a b + 35 b f + 8 a h + 10 f h + 12 a x + 15 f x \][/tex]
So, the simplified answer is:
[tex]\[ 20 a m + 25 f m + 28 a b + 35 b f + 8 a h + 10 f h + 12 a x + 15 f x \][/tex]
2. Combining Like Terms in the Polynomial:
[tex]\[ 8 x^3 y^3 + 48 y^4 x^5 + 96 x^7 y^5 + 64 x^9 y^6 \][/tex]
Here, there are no similar terms to combine. Therefore, the expression remains as:
[tex]\[ 8 x^3 y^3 + 48 y^4 x^5 + 96 x^7 y^5 + 64 x^9 y^6 \][/tex]
3. Simplifying the Fraction and Expression:
[tex]\[ \frac{4}{64} b^{10} y^8 - 0.81 b^4 c^6 \][/tex]
First, simplify the fractional coefficient:
[tex]\[ \frac{4}{64} = \frac{1}{16} \][/tex]
So, the expression becomes:
[tex]\[ \frac{1}{16} b^{10} y^8 - 0.81 b^4 c^6 \][/tex]
4. Simplifying the Cubic Expression:
[tex]\[ c^3 - 3^3 \][/tex]
Recall that:
[tex]\[ 3^3 = 27 \][/tex]
Thus, the simplified expression is:
[tex]\[ c^3 - 27 \][/tex]
5. Factoring the Polynomial Expression:
[tex]\[ -27 a^{12} b^6 - 9 a^{12} b^4 - a^{12} b^3 - 27 a^{12} b^5 \][/tex]
Factor out the common factor [tex]\(-a^{12}\)[/tex]:
[tex]\[ -a^{12}(27 b^6 + 9 b^4 + b^3 + 27 b^5) \][/tex]
Rearrange the terms inside the parentheses:
[tex]\[ -a^{12}(27 b^6 + 27 b^5 + 9 b^4 + b^3) \][/tex]
Factor out 27 from the first two terms:
[tex]\[ -a^{12} \times 27(b^6 + b^5) - a^{12}(9 b^4 + b^3) \][/tex]
Combine all the terms under the common factor [tex]\(-64 a^{12}\)[/tex]:
[tex]\[ -64 a^{12}(b^6 + b^5 + b^4 + b^3) \][/tex]
So, the simplified answer is:
[tex]\[ -64 a^{12}(b^6 + b^5 + b^4 + b^3) \][/tex]
### Summary:
1. [tex]\( 20 a m + 25 f m + 28 a b + 35 b f + 8 a h + 10 f h + 12 a x + 15 f x \)[/tex]
2. [tex]\( 8 x^3 y^3 + 48 y^4 x^5 + 96 x^7 y^5 + 64 x^9 y^6 \)[/tex]
3. [tex]\( \frac{1}{16} b^{10} y^8 - 0.81 b^4 c^6 \)[/tex]
4. [tex]\( c^3 - 27 \)[/tex]
5. [tex]\( -64 a^{12}(b^6 + b^5 + b^4 + b^3) \)[/tex]
Buena Suerte!!
