11. $\frac{16x^6 - 12x^4 + 4x^2}{4x^2}$\
12. $\frac{12c^3d^4 + 18c^4d^3}{3c^2d^3}$\
13. $\frac{-24x^7 + 9x^3 - 15x}{3x^5}$\
14. $\frac{15x^5 - 25x^3 + 5x^2}{5x^4}$\
15. $\frac{28x^5y^4z^3 + 8x^4y^3z^2}{4x^2y^2z^2}$\
16. $\frac{30r^5s^9 - 12r^4s^8 + 3r^3s^7}{3r^2s^2}$\
directions: simplify each expression completely.\
17. $\frac{15x^2 + 6x}{3x} \cdot (2x - 1)$\
18. $\frac{2r^3s^8(7r^7 - 5r^3s^4) - (2r^{10}s^8 - 6r^6s^{12})}{4r^2s^4}$\
19. $\frac{(m^2n^2 - mn)(m^2n^2 + mn)}{mn}$

Question

Accepted Answer

### Problem 11: $\boldsymbol{\frac{16x^5 - 12x^4 + 4x^2}{4x^2}}$
# Explanation:
## Step 1: Divide each term by $4x^2$
We use the rule $\frac{a + b + c}{d}=\frac{a}{d}+\frac{b}{d}+\frac{c}{d}$. So we have:
$\frac{16x^5}{4x^2}-\frac{12x^4}{4x^2}+\frac{4x^2}{4x^2}$

## Step 2: Simplify each term
For the first term: $\frac{16x^5}{4x^2}=\frac{16}{4}	imes x^{5 - 2}=4x^3$
For the second term: $\frac{12x^4}{4x^2}=\frac{12}{4}	imes x^{4 - 2}=3x^2$
For the third term: $\frac{4x^2}{4x^2} = 1$

# Answer:
$4x^3-3x^2 + 1$

### Problem 12: $\boldsymbol{\frac{12c^3d^4+18c^4d^3}{3c^2d^3}}$
# Explanation:
## Step 1: Divide each term by $3c^2d^3$
Using the rule $\frac{a + b}{d}=\frac{a}{d}+\frac{b}{d}$, we get:
$\frac{12c^3d^4}{3c^2d^3}+\frac{18c^4d^3}{3c^2d^3}$

## Step 2: Simplify each term
First term: $\frac{12c^3d^4}{3c^2d^3}=\frac{12}{3}	imes c^{3 - 2}	imes d^{4 - 3}=4cd$
Second term: $\frac{18c^4d^3}{3c^2d^3}=\frac{18}{3}	imes c^{4 - 2}	imes d^{3 - 3}=6c^2$

# Answer:
$4cd + 6c^2$

### Problem 13: $\boldsymbol{\frac{-24x^7+9x^3 - 15x}{3x^5}}$
# Explanation:
## Step 1: Divide each term by $3x^5$
Using the rule $\frac{a + b + c}{d}=\frac{a}{d}+\frac{b}{d}+\frac{c}{d}$, we have:
$\frac{-24x^7}{3x^5}+\frac{9x^3}{3x^5}-\frac{15x}{3x^5}$

## Step 2: Simplify each term
First term: $\frac{-24x^7}{3x^5}=\frac{-24}{3}	imes x^{7 - 5}=-8x^2$
Second term: $\frac{9x^3}{3x^5}=\frac{9}{3}	imes x^{3 - 5}=3x^{-2}=\frac{3}{x^2}$
Third term: $\frac{-15x}{3x^5}=\frac{-15}{3}	imes x^{1 - 5}=-5x^{-4}=\frac{-5}{x^4}$

# Answer:
$-8x^2+\frac{3}{x^2}-\frac{5}{x^4}$

### Problem 14: $\boldsymbol{\frac{15x^5-25x^3 + 5x^2}{5x^4}}$
# Explanation:
## Step 1: Divide each term by $5x^4$
Using the rule $\frac{a + b + c}{d}=\frac{a}{d}+\frac{b}{d}+\frac{c}{d}$, we get:
$\frac{15x^5}{5x^4}-\frac{25x^3}{5x^4}+\frac{5x^2}{5x^4}$

## Step 2: Simplify each term
First term: $\frac{15x^5}{5x^4}=\frac{15}{5}	imes x^{5 - 4}=3x$
Second term: $\frac{-25x^3}{5x^4}=\frac{-25}{5}	imes x^{3 - 4}=-5x^{-1}=\frac{-5}{x}$
Third term: $\frac{5x^2}{5x^4}=\frac{5}{5}	imes x^{2 - 4}=x^{-2}=\frac{1}{x^2}$

# Answer:
$3x-\frac{5}{x}+\frac{1}{x^2}$

### Problem 15: $\boldsymbol{\frac{28x^5y^4z^3+8x^4y^3z^2}{4x^2y^2z^2}}$
# Explanation:
## Step 1: Divide each term by $4x^2y^2z^2$
Using the rule $\frac{a + b}{d}=\frac{a}{d}+\frac{b}{d}$, we have:
$\frac{28x^5y^4z^3}{4x^2y^2z^2}+\frac{8x^4y^3z^2}{4x^2y^2z^2}$

## Step 2: Simplify each term
First term: $\frac{28x^5y^4z^3}{4x^2y^2z^2}=\frac{28}{4}	imes x^{5 - 2}	imes y^{4 - 2}	imes z^{3 - 2}=7x^3y^2z$
Second term: $\frac{8x^4y^3z^2}{4x^2y^2z^2}=\frac{8}{4}	imes x^{4 - 2}	imes y^{3 - 2}	imes z^{2 - 2}=2x^2y$

# Answer:
$7x^3y^2z + 2x^2y$

### Problem 16: $\boldsymbol{\frac{30r^5s^9-12r^4s^8 + 3r^3s^7}{3r^2s^2}}$
# Explanation:
## Step 1: Divide each term by $3r^2s^2$
Using the rule $\frac{a + b + c}{d}=\frac{a}{d}+\frac{b}{d}+\frac{c}{d}$, we get:
$\frac{30r^5s^9}{3r^2s^2}-\frac{12r^4s^8}{3r^2s^2}+\frac{3r^3s^7}{3r^2s^2}$

## Step 2: Simplify each term
First term: $\frac{30r^5s^9}{3r^2s^2}=\frac{30}{3}	imes r^{5 - 2}	imes s^{9 - 2}=10r^3s^7$
Second term: $\frac{-12r^4s^8}{3r^2s^2}=\frac{-12}{3}	imes r^{4 - 2}	imes s^{8 - 2}=-4r^2s^6$
Third term: $\frac{3r^3s^7}{3r^2s^2}=\frac{3}{3}	imes r^{3 - 2}	imes s^{7 - 2}=rs^5$

# Answer:
$10r^3s^7-4r^2s^6 + rs^5$

### Problem 17: $\boldsymbol{\frac{15x^2+6x}{3x}\cdot(2x - 1)}$
# Explanation:
## Step 1: Simplify the fraction
Simplify $\frac{15x^2+6x}{3x}$ by dividing each term in the numerator by $3x$:
$\frac{15x^2}{3x}+\frac{6x}{3x}=5x + 2$

## Step 2: Multiply by $(2x - 1)$
Now multiply $(5x + 2)$ with $(2x - 1)$ using the distributive property $(a + b)(c + d)=ac+ad+bc+bd$:
$(5x)(2x)+(5x)(-1)+(2)(2x)+(2)(-1)$
$=10x^2-5x + 4x-2$

## Step 3: Combine like terms
Combine $-5x$ and $4x$:
$10x^2 - x-2$

# Answer:
$10x^2 - x - 2$

### Problem 18: $\boldsymbol{\frac{2r^3s^8(7r^7-5r^3s^4)-(2r^{10}s^8-6r^6s^{12})}{4r^2s^4}}$
# Explanation:
## Step 1: Expand the numerator
First, expand $2r^3s^8(7r^7-5r^3s^4)$ using the distributive property:
$2r^3s^8	imes7r^7-2r^3s^8	imes5r^3s^4=14r^{10}s^8-10r^6s^{12}$
Then subtract $(2r^{10}s^8-6r^6s^{12})$ from the above result:
$(14r^{10}s^8-10r^6s^{12})-(2r^{10}s^8-6r^6s^{12})$
$=14r^{10}s^8-10r^6s^{12}-2r^{10}s^8 + 6r^6s^{12}$

## Step 2: Combine like terms in the numerator
Combine $14r^{10}s^8$ and $-2r^{10}s^8$: $12r^{10}s^8$
Combine $-10r^6s^{12}$ and $6r^6s^{12}$: $-4r^6s^{12}$
So the numerator becomes $12r^{10}s^8-4r^6s^{12}$

## Step 3: Divide each term in the numerator by $4r^2s^4$
Divide $12r^{10}s^8$ by $4r^2s^4$: $\frac{12r^{10}s^8}{4r^2s^4}=3r^{8}s^4$
Divide $-4r^6s^{12}$ by $4r^2s^4$: $\frac{-4r^6s^{12}}{4r^2s^4}=-r^4s^8$

# Answer:
$3r^8s^4-r^4s^8$

### Problem 19: $\boldsymbol{\frac{(m^2n^2-mn)(m^2n^2+mn)}{mn}}$
# Explanation:
## Step 1: Simplify the numerator using difference of squares
The numerator $(m^2n^2-mn)(m^2n^2+mn)$ is in the form $(a - b)(a + b)=a^2 - b^2$ where $a = m^2n^2$ and $b = mn$. So:
$(m^2n^2)^2-(mn)^2=m^4n^4 - m^2n^2$

## Step 2: Divide by $mn$
Now divide $m^4n^4 - m^2n^2$ by $mn$ by dividing each term:
$\frac{m^4n^4}{mn}-\frac{m^2n^2}{mn}$

## Step 3: Simplify each term
First term: $\frac{m^4n^4}{mn}=m^{3}n^{3}$
Second term: $\frac{-m^2n^2}{mn}=-mn$

# Answer:
$m^3n^3 - mn$

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QUESTION IMAGE

Question

Explanation:

Problem 11: $\boldsymbol{\frac{16x^5 - 12x^4 + 4x^2}{4x^2}}$

Step 1: Divide each term by $4x^2$

Step 2: Simplify each term

Step 1: Divide each term by $3c^2d^3$

Step 2: Simplify each term

Step 1: Divide each term by $3x^5$

Step 2: Simplify each term

Answer:

Problem 12: $\boldsymbol{\frac{12c^3d^4+18c^4d^3}{3c^2d^3}}$