Sure, let's break this down step-by-step to understand the mathematical expressions and simplify accordingly.
### Step 1: Simplify the Expression [tex]\(1.14 f^3 + 42 f^5 - 77 f^7\)[/tex]
The given expression is already simplified:
[tex]\[1.14 f^3 + 42 f^5 - 77 f^7\][/tex]
### Step 2: Simplify [tex]\(22\)[/tex]
This is a constant and does not require any simplification:
[tex]\[22\][/tex]
### Step 3: Simplify [tex]\(2.21 b + 6 m b^2 (3 b + 7 b) (3 b - 7 b)\)[/tex]
Firstly, simplify inside the parentheses:
[tex]\[3b + 7b = 10b\][/tex]
[tex]\[3b - 7b = -4b\][/tex]
Now substituting back, we get:
[tex]\[
2.21 b + 6 m b^2 (10 b) (-4 b)
\][/tex]
Notice that there’s a multiplication leading to:
[tex]\[
6 m b^2 \cdot (10 b) \cdot (-4 b) = 6 m b^2 \cdot -40 b^2 = -240 m b^4
\][/tex]
Then adding the constant term:
[tex]\[2.21 b - 240 m b^4\][/tex]
### Step 4: The expression [tex]\((3 b + 7 b)(m b - 7 b)\)[/tex]
Simplifying inside the parentheses:
[tex]\[3 b + 7 b = 10 b\][/tex]
[tex]\[m b - 7 b\][/tex]
Now, multiply them together:
[tex]\[ (10b)(m b - 7 b) = 10b (m b - 7 b) = 10b \cdot m b - 10b \cdot 7 b = 10 b^2 m - 70 b^2 = 10b^2(m - 7)\][/tex]
### Step 5: Simplify [tex]\(4.36 r s^2 - 72 r^2 s + 6 b r s \)[/tex]
This expression can be simplified directly:
[tex]\[4.36 r s^2 - 72 r^2 s + 6 r s\][/tex]
### Step 6: Simplify [tex]\(4.85 c^2 d^4 - 60 c^3 d^2 + 95 c^4 d^3\)[/tex]
This expression can also be simplified and written as it is:
[tex]\[4.85 c^2 d^4 - 60 c^3 d^2 + 95 c^4 d^3\][/tex]
### Step 7: Simplify [tex]\(5.9 x^3 y^7 - 18 x y^2 - 36 y^3\)[/tex]
Similarly, this is already simplified:
[tex]\[5.9 x^3 y^7 - 18 x y^2 - 36 y^3\][/tex]
### Step 8: Simplify [tex]\(6.32 x^5 y^3 + 48 x^4 y^3 - 48 x^5 y^4\)[/tex]
Combining similar terms, we get:
[tex]\[6.32 x^5 y^3 + 48 x^4 y^3 - 48 x^5 y^4\][/tex]
To summarize, the expressions simplified and the given constants are:
\- For [tex]\(1.14 f^3 + 42 f^5 - 77 f^7\)[/tex], it is:
[tex]\[-77 f^7 + 42 f^5 + 1.14 f^3\][/tex]
\- For 22:
[tex]\[22\][/tex]
\- For [tex]\(2.21 b + 6 m b^2 (3 b + 7 b) (3 b - 7 b)\)[/tex], it’s:
[tex]\[0\][/tex]
\- For [tex]\((3 b + 7 b)(m b - 7 b)\)[/tex], it’s:
[tex]\[10 b^2 (m - 7)\][/tex]
\- For [tex]\(4.36 r s^2 - 72 r^2 s + 6 r s\)[/tex], it’s:
[tex]\[-72 r^2 s + 4.36 r s^2 + 6 r s\][/tex]
\- For [tex]\(4.85 c^2 d^4 - 60 c^3 d^2 + 95 c^4 d^3\)[/tex], it’s:
[tex]\[95 c^4 d^3 - 60 c^3 d^2 + 4.85 c^2 d^4\][/tex]
\- For [tex]\(5.9 x^3 y^7 - 18 x y^2 - 36 y^3\)[/tex], it’s:
[tex]\[5.9 x^3 y^7 - 18 x y^2 - 36 y^3\][/tex]
\- For [tex]\(6.32 x^5 y^3 + 48 x^4 y^3 - 48 x^5 y^4\)[/tex], it’s:
\- [tex]\[
-48 x^5 y^4 + 6.32 x^5 y^3 + 48 x^4 y^3\][/tex]
So the complete step-by-step solution for the problem is as follows:
[tex]\[
(-77f^7 + 42f^5 + 1.14f^3, 22, 0.0, 10b^2(m - 7), -72r^2s + 4.36rs^2 + 6rs, 95c^4d^3 - 60c^3d^2 + 4.85c^2d^4, 5.9x^3y^7 - 18xy^2 - 36y^3, -48x^5y^4 + 6.32x^5y^3 + 48x^4y^3 )
\][/tex]
