Question

topic 1: algebra 2 review
monomials and polynomials
exponent rules
product rule
$x^a \cdot x^b =$
power rule
$(x^a)^b =$
quotient rule
$\frac{x^a}{x^b} =$
negative exponent
rules
$x^{-a} =$
what is the rule for adding and subtracting monomials?
simplify each expression completely. final answer should contain only positive exponents.
1.
$6ab - 8ab$
$-2ab$
2.
$-2xy^2 - 4xy + 6xy^2$
$4xy^2 - 4xy$
3.
subtract $-6b^5$ from $8b^5$.
$14b^5$
4.
$-7n^{-4} \cdot 5n^{-2}$
$-35n^{-6}$ $\frac{1}{35n^6}$
5.
$(5v^4)^2 \cdot 2v^3 \cdot v$
$25v^8 \cdot 2v^5 \cdot v = 50v^{12}$
6.
$(-a^6b)^2 + 9a^{12}b^2$
7.
$(-2y^4) \cdot (xy^3)^2 - 13x^2y^{10}$
$(-2y^4) \cdot x^2y^6 = -2x^2y^{10}$
$-2x^2y^{10} - 13x^2y^{10} = -15x^2y^{10}$
8.
$\frac{r^6s^7t^2}{r^5s^4t^2}$
$r^1s^3$
9.
$\frac{(-3x^6)^2}{5x^3 \cdot 3x^3}$

Explanation:

Step1: Define Product Rule

$x^a \cdot x^b = x^{a+b}$

Step2: Define Power Rule

$(x^a)^b = x^{a \cdot b}$

Step3: Define Quotient Rule

$\frac{x^a}{x^b} = x^{a-b}$

Step4: Define Negative Exponent Rule

$x^{-a} = \frac{1}{x^a}$

Step5: State monomial addition/subtraction rule

Only like monomials (same variables with identical exponents) can be added or subtracted by combining their coefficients.
---

Step6: Simplify $6ab-8ab$

Combine like terms: $(6-8)ab = -2ab$

Step7: Simplify $-2xy^2-4xy+6xy^2$

Combine like $xy^2$ terms: $(-2+6)xy^2 -4xy = 4xy^2 -4xy$

Step8: Subtract $-6b^5$ from $8b^5$

Rewrite as $8b^5 - (-6b^5) = (8+6)b^5 = 14b^5$

Step9: Simplify $-7n^{-4} \cdot 5n^{-2}$

Multiply coefficients, add exponents: $(-7 \cdot 5)n^{-4-2} = -35n^{-6} = \frac{-35}{n^6}$

Step10: Simplify $(5v^4)^2 \cdot 2v^3 \cdot v$

Apply power rule, then product rule: $25v^8 \cdot 2v^{3+1} = 25v^8 \cdot 2v^4 = 50v^{12}$

Step11: Simplify $(-a^6b)^2 + 9a^{12}b^2$

Apply power rule, combine terms: $a^{12}b^2 + 9a^{12}b^2 = 10a^{12}b^2$

Step12: Simplify $(-2y^4) \cdot (xy^3)^2 -13x^2y^{10}$

Apply power rule, multiply, combine terms: $(-2y^4)(x^2y^6) -13x^2y^{10} = -2x^2y^{10} -13x^2y^{10} = -15x^2y^{10}$

Step13: Simplify $\frac{r^6s^7t^2}{r^5s^4t^2}$

Apply quotient rule: $r^{6-5}s^{7-4}t^{2-2} = r s^3$

Step14: Simplify $\frac{(-3x^6)^2}{5x^3 \cdot 3x^3}$

Apply power rule, product rule, quotient rule: $\frac{9x^{12}}{15x^{6}} = \frac{3}{5}x^{6}$

Answer:

Exponent Rules:

• Product Rule: $x^a \cdot x^b = x^{a+b}$
• Power Rule: $(x^a)^b = x^{ab}$
• Quotient Rule: $\frac{x^a}{x^b} = x^{a-b}$
• Negative Exponent Rule: $x^{-a} = \frac{1}{x^a}$

Monomial Addition/Subtraction Rule:

Only like monomials (same variables with matching exponents) can be combined by adding or subtracting their coefficients.

Simplified Expressions:

1. $-2ab$
2. $4xy^2 - 4xy$
3. $14b^5$
4. $\frac{-35}{n^6}$
5. $50v^{12}$
6. $10a^{12}b^2$
7. $-15x^2y^{10}$
8. $rs^3$
9. $\frac{3}{5}x^6$