term in the denominator. for exponents, recall the quotient rule.
example 1: $\frac{8n^2 - 36n}{4n}$
example 2: $\frac{-14x^{10}y^3 + x^7y^3}{x^7y^3}$
directions: find each quotient.
1. $\frac{6x + 9}{3}$
2. $\frac{10v^3 + 5v - 15}{5}$
3. $\frac{8a^3 - 2a^2 + 5a}{a}$
4. $\frac{40x^3 - 8x}{8x}$
5. $\frac{18c^3 - 21c^2 + 3c}{3c}$
6. $\frac{6a^2b - 12ab^3}{-2ab}$
7. $\frac{-20x^2 - 5x}{5x}$
8. $\frac{14x^6y^3 - 49x^3y^3}{-7x^3y}$
9. $\frac{-25x^4y^3 + 30x^2y^5}{-5x^2y}$
10. $\frac{20a^7b^3c^3 - 5abc}{5abc}$

Question

Accepted Answer

Let's solve problem 1: $\boldsymbol{\frac{6x + 9}{3}}$

# Explanation:
## Step 1: Split the fraction
We can split the fraction into two separate fractions: $\frac{6x}{3} + \frac{9}{3}$
## Step 2: Simplify each fraction
Simplify $\frac{6x}{3}$: $6x \div 3 = 2x$
Simplify $\frac{9}{3}$: $9 \div 3 = 3$

# Answer:
$2x + 3$

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Let's solve problem 2: $\boldsymbol{\frac{10v^3 + 5v - 15}{5}}$

# Explanation:
## Step 1: Split the fraction
Split into three fractions: $\frac{10v^3}{5} + \frac{5v}{5} - \frac{15}{5}$
## Step 2: Simplify each fraction
Simplify $\frac{10v^3}{5}$: $10v^3 \div 5 = 2v^3$
Simplify $\frac{5v}{5}$: $5v \div 5 = v$
Simplify $\frac{15}{5}$: $15 \div 5 = 3$

# Answer:
$2v^3 + v - 3$

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Let's solve problem 3: $\boldsymbol{\frac{8a^3 - 2a^2 + 5a}{a}}$

# Explanation:
## Step 1: Split the fraction
Split into three fractions: $\frac{8a^3}{a} - \frac{2a^2}{a} + \frac{5a}{a}$
## Step 2: Simplify each fraction
Simplify $\frac{8a^3}{a}$: $8a^3 \div a = 8a^2$ (using $a^m \div a^n = a^{m-n}$)
Simplify $\frac{2a^2}{a}$: $2a^2 \div a = 2a$
Simplify $\frac{5a}{a}$: $5a \div a = 5$

# Answer:
$8a^2 - 2a + 5$

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Let's solve problem 4: $\boldsymbol{\frac{40x^3 - 8x}{8x}}$

# Explanation:
## Step 1: Split the fraction
Split into two fractions: $\frac{40x^3}{8x} - \frac{8x}{8x}$
## Step 2: Simplify each fraction
Simplify $\frac{40x^3}{8x}$: $40 \div 8 = 5$ and $x^3 \div x = x^2$, so $5x^2$
Simplify $\frac{8x}{8x}$: $8x \div 8x = 1$

# Answer:
$5x^2 - 1$

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Let's solve problem 5: $\boldsymbol{\frac{18c^3 - 21c^2 + 3c}{3c}}$

# Explanation:
## Step 1: Split the fraction
Split into three fractions: $\frac{18c^3}{3c} - \frac{21c^2}{3c} + \frac{3c}{3c}$
## Step 2: Simplify each fraction
Simplify $\frac{18c^3}{3c}$: $18 \div 3 = 6$ and $c^3 \div c = c^2$, so $6c^2$
Simplify $\frac{21c^2}{3c}$: $21 \div 3 = 7$ and $c^2 \div c = c$, so $7c$
Simplify $\frac{3c}{3c}$: $3c \div 3c = 1$

# Answer:
$6c^2 - 7c + 1$ (Note: The handwritten answer in the image has a typo; the correct constant term is 1, not 3)

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Let's solve problem 6: $\boldsymbol{\frac{6a^2b - 12ab^3}{-2ab}}$

# Explanation:
## Step 1: Split the fraction
Split into two fractions: $\frac{6a^2b}{-2ab} - \frac{12ab^3}{-2ab}$
## Step 2: Simplify each fraction
Simplify $\frac{6a^2b}{-2ab}$: $6 \div (-2) = -3$ and $a^2 \div a = a$, $b \div b = 1$, so $-3a$
Simplify $\frac{12ab^3}{-2ab}$: $12 \div (-2) = -6$ and $a \div a = 1$, $b^3 \div b = b^2$, so $6b^2$ (negative divided by negative is positive)

# Answer:
$-3a + 6b^2$ (or $6b^2 - 3a$)

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Let's solve problem 7: $\boldsymbol{\frac{-20x^2 - 5x}{5x}}$

# Explanation:
## Step 1: Split the fraction
Split into two fractions: $\frac{-20x^2}{5x} - \frac{5x}{5x}$
## Step 2: Simplify each fraction
Simplify $\frac{-20x^2}{5x}$: $-20 \div 5 = -4$ and $x^2 \div x = x$, so $-4x$
Simplify $\frac{5x}{5x}$: $5x \div 5x = 1$

# Answer:
$-4x - 1$

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Let's solve problem 8: $\boldsymbol{\frac{14x^6y^3 - 49x^3y^3}{-7x^4y}}$

# Explanation:
## Step 1: Split the fraction
Split into two fractions: $\frac{14x^6y^3}{-7x^4y} - \frac{49x^3y^3}{-7x^4y}$
## Step 2: Simplify each fraction
Simplify $\frac{14x^6y^3}{-7x^4y}$: $14 \div (-7) = -2$, $x^6 \div x^4 = x^2$, $y^3 \div y = y^2$, so $-2x^2y^2$
Simplify $\frac{49x^3y^3}{-7x^4y}$: $49 \div (-7) = -7$, $x^3 \div x^4 = \frac{1}{x}$, $y^3 \div y = y^2$, so $\frac{7y^2}{x}$ (negative divided by negative is positive)

# Answer:
$-2x^2y^2 + \frac{7y^2}{x}$

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Let's solve problem 9: $\boldsymbol{\frac{-25x^4y^3 + 30x^2y^5}{-5x^2y}}$

# Explanation:
## Step 1: Split the fraction
Split into two fractions: $\frac{-25x^4y^3}{-5x^2y} + \frac{30x^2y^5}{-5x^2y}$
## Step 2: Simplify each fraction
Simplify $\frac{-25x^4y^3}{-5x^2y}$: $-25 \div (-5) = 5$, $x^4 \div x^2 = x^2$, $y^3 \div y = y^2$, so $5x^2y^2$
Simplify $\frac{30x^2y^5}{-5x^2y}$: $30 \div (-5) = -6$, $x^2 \div x^2 = 1$, $y^5 \div y = y^4$, so $-6y^4$

# Answer:
$5x^2y^2 - 6y^4$

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Let's solve problem 10: $\boldsymbol{\frac{20a^7b^3c^3 - 5abc}{5abc}}$

# Explanation:
## Step 1: Split the fraction
Split into two fractions: $\frac{20a^7b^3c^3}{5abc} - \frac{5abc}{5abc}$
## Step 2: Simplify each fraction
Simplify $\frac{20a^7b^3c^3}{5abc}$: $20 \div 5 = 4$, $a^7 \div a = a^6$, $b^3 \div b = b^2$, $c^3 \div c = c^2$, so $4a^6b^2c^2$
Simplify $\frac{5abc}{5abc}$: $5abc \div 5abc = 1$

# Answer:
$4a^6b^2c^2 - 1$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step 1: Split the fraction

Step 2: Simplify each fraction

Step 1: Split the fraction

Step 2: Simplify each fraction

Step 1: Split the fraction

Step 2: Simplify each fraction

Answer: