Question

algebra 2 ws 4.3 (day 1) dividing expressions name tyler boettcher hr 8
problems 1 - 5: simplify each expression, if possible.
circle \true\ if the expression is simplified correctly. circle \false\ if it is not.

1. true false \\(\frac{2xy - 4}{6} = \frac{xy}{3} - \frac{2}{3}\\)
2. true false \\(\frac{6}{2xy - 4} = \frac{3}{xy} - \frac{3}{2}\\)
3. true false \\(\frac{m^2n - 5m}{10mn} = 10m - 2n\\)
4. true false \\(\frac{2x - 14 + 5y}{\frac{1}{2}} = 4x - 28 + 10y\\)
5. true false \\(\frac{4n + 3n^3 + 12n^4p}{12n} = \frac{1}{3} + \frac{n^2}{4} + n^3\\)

problems 6 - 8: simplify the expressions.

1. \\(\frac{8x^5y + 12x^3y^2}{6x^4y^2}\\)
2. \\(\frac{5a^2 + 2a - 1}{\frac{1}{2}}\\)
3. \\(\frac{30m^6n^5 + 40m^9 - 50n^8}{20m^5n^5}\\)

Explanation:

Problems 1-5:

Step1: Simplify left-hand side (LHS)

$\frac{2xy - 4}{6} = \frac{2xy}{6} - \frac{4}{6} = \frac{xy}{3} - \frac{2}{3}$

Step2: Compare to right-hand side (RHS)

LHS matches $\frac{xy}{3} - \frac{2}{3}$, so the statement is true.

Step1: Simplify LHS

$\frac{6}{2xy - 4} = \frac{6}{2(xy - 2)} = \frac{3}{xy - 2}$

Step2: Compare to RHS

$\frac{3}{xy - 2}
eq \frac{3}{xy} - \frac{3}{2}$, so the statement is false.

Step1: Simplify LHS

$\frac{m^2n - 5m}{10mn} = \frac{m^2n}{10mn} - \frac{5m}{10mn} = \frac{m}{10} - \frac{1}{2n}$

Step2: Compare to RHS

$\frac{m}{10} - \frac{1}{2n}
eq 10m - 2n$, so the statement is false.

Step1: Simplify LHS

$\frac{2x - 14 + 5y}{\frac{1}{2}} = (2x - 14 + 5y) \times 2 = 4x - 28 + 10y$

Step2: Compare to RHS

LHS matches $4x - 28 + 10y$, so the statement is true.

Step1: Simplify LHS

$\frac{4n + 3n^3 + 12n^4p}{12n} = \frac{4n}{12n} + \frac{3n^3}{12n} + \frac{12n^4p}{12n} = \frac{1}{3} + \frac{n^2}{4} + n^3p$

Step2: Compare to RHS

$\frac{1}{3} + \frac{n^2}{4} + n^3p
eq \frac{1}{3} + \frac{n^2}{4} + n^3$, so the statement is false.

Problem 6 Step1: Split the fraction

$\frac{8x^5y + 12x^3y^3}{6x^4y^2} = \frac{8x^5y}{6x^4y^2} + \frac{12x^3y^3}{6x^4y^2}$

Problem 6 Step2: Simplify each term

$\frac{8x^5y}{6x^4y^2} = \frac{4x}{3y}$, $\frac{12x^3y^3}{6x^4y^2} = \frac{2y}{x}$
$\text{Combined: } \frac{4x}{3y} + \frac{2y}{x}$

Problem 7 Step1: Rewrite division as multiplication

$\frac{5a^2 + 2a - 1}{\frac{1}{2}} = (5a^2 + 2a - 1) \times 2$

Problem 7 Step2: Distribute the multiplier

$2 \times 5a^2 + 2 \times 2a - 2 \times 1 = 10a^2 + 4a - 2$

Problem 8 Step1: Split the fraction

$\frac{30m^6n^5 + 40m^9 - 50n^8}{20m^5n^5} = \frac{30m^6n^5}{20m^5n^5} + \frac{40m^9}{20m^5n^5} - \frac{50n^8}{20m^5n^5}$

Problem 8 Step2: Simplify each term

$\frac{30m^6n^5}{20m^5n^5} = \frac{3m}{2}$, $\frac{40m^9}{20m^5n^5} = \frac{2m^4}{n^5}$, $\frac{50n^8}{20m^5n^5} = \frac{5n^3}{2m^5}$
$\text{Combined: } \frac{3m}{2} + \frac{2m^4}{n^5} - \frac{5n^3}{2m^5}$

Answer:

(Problems 1-5):

1. true
2. false
3. false
4. true
5. false

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Problems 6-8: