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>Parentheses>Activities 1

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Activities 1

1.1)

Find the answers and take note of what the parentheses do to the calculation and the answer:

a): $7 \times 4 + 3$
b): $7 \times (4 + 3)$
c): $10 - 3 \times 3$
d): $(10 - 3) \times 3$
e): $5 \times 8 + 4 \times 9$
f): $5 \times (8 + 4) \times 9$
g): $20 + 3 \times 7 - 4 \times 2 + 9$
h): $((20 + 3) \times 7) - 4 \times (2 + 9)$

a): $7 \times 4 + 3$
b): $7 \times (4 + 3)$
c): $10 - 3 \times 3$
d): $(10 - 3) \times 3$
e): $5 \times 8 + 4 \times 9$
f): $5 \times (8 + 4) \times 9$
g): $20 + 3 \times 7 - 4 \times 2 + 9$
h): $((20 + 3) \times 7) - 4 \times (2 + 9)$

1.2)

Expand the parentheses and find the answers:

a): $(8 + 19) + (12 - 4)$
b): $- (4 + 13) + (3 + 26)$
c): $(7 - 12) - (9 - 16)$
d): $- (- 8 + 3) + (- 7 + 20)$
e): $(- 10 + 8) + (- 5 - 3)$
f): $- (100 - 20) + (- 5 + (- 12))$
g): $(6 - 5) - (7 - (- 14))$
h): $- (- 4 - 3) - (- 9 - (- 5))$

a): $(8 + 19) + (12 - 4)$
b): $- (4 + 13) + (3 + 26)$
c): $(7 - 12) - (9 - 16)$
d): $- (- 8 + 3) + (- 7 + 20)$
e): $(- 10 + 8) + (- 5 - 3)$
f): $- (100 - 20) + (- 5 + (- 12))$
g): $(6 - 5) - (7 - (- 14))$
h): $- (- 4 - 3) - (- 9 - (- 5))$

1.3)

Find the answers:

a): $3 - 1 \times 3$
b): $(3 - 1) \times 3$
c): $(4 - 2) \times (2 + 5)$
d): $2 (3 + 2)$
e): $14 (2 + 1)$
f): $3 \times 2 + 4 \times 2 - 3 \times 5$
g): $(3 + 4) (11 + 19)$
h): $7 (2 - 9)$
i): $17 + 3 (2 + 5) - 2 (1 + (- 8))$

a): $3 - 1 \times 3$
b): $(3 - 1) \times 3$
c): $(4 - 2) \times (2 + 5)$
d): $2 (3 + 2)$
e): $14 (2 + 1)$
f): $3 \times 2 + 4 \times 2 - 3 \times 5$
g): $(3 + 4) (11 + 19)$
h): $7 (2 - 9)$
i): $17 + 3 (2 + 5) - 2 (1 + (- 8))$

1.4)

Expand the expressions and simplify them as much as possible:

a): $2 \times (\frac{3}{8} + \frac{1}{4})$
b): $(\frac{5}{6} - \frac{2}{9}) \times \frac{3}{5}$
c): $(\frac{5}{36} + \frac{1}{12}) \frac{2}{9}$
d): $(\frac{7}{6} - \frac{2}{9}) \times (\frac{1}{5} + \frac{1}{4})$
e): $\frac{4}{33} (32 - 26)$
f): $(29 - 19) \frac{1}{60}$
g): $(4 + \frac{9}{4}) \frac{4}{5}$
h): $\frac{8}{3} (\frac{9}{5} + 7)$

a): $2 \times (\frac{3}{8} + \frac{1}{4})$
b): $(\frac{5}{6} - \frac{2}{9}) \times \frac{3}{5}$
c): $(\frac{5}{36} + \frac{1}{12}) \frac{2}{9}$
d): $(\frac{7}{6} - \frac{2}{9}) \times (\frac{1}{5} + \frac{1}{4})$
e): $\frac{4}{33} (32 - 26)$
f): $(29 - 19) \frac{1}{60}$
g): $(4 + \frac{9}{4}) \frac{4}{5}$
h): $\frac{8}{3} (\frac{9}{5} + 7)$

1.5)

Expand the expressions and simplify them as much as possible:

a): $3 (\frac{1}{2} + \frac{3}{4})$
b): $12 (\frac{1}{3} + \frac{1}{4})$
c): $24 (\frac{2}{8} + \frac{3}{6})$
d): $\frac{1}{2} (\frac{5}{9} + \frac{24}{10})$
e): $4 (\frac{2}{5} - \frac{4}{7})$
f): $(\frac{6}{10} - \frac{1}{3}) 5$
g): $(4 - \frac{1}{2}) 7$
h): $(\frac{6}{5} + 2) 8$

a): $3 (\frac{1}{2} + \frac{3}{4})$
b): $12 (\frac{1}{3} + \frac{1}{4})$
c): $24 (\frac{2}{8} + \frac{3}{6})$
d): $\frac{1}{2} (\frac{5}{9} + \frac{24}{10})$
e): $4 (\frac{2}{5} - \frac{4}{7})$
f): $(\frac{6}{10} - \frac{1}{3}) 5$
g): $(4 - \frac{1}{2}) 7$
h): $(\frac{6}{5} + 2) 8$

1.6)

Expand the expressions and simplify them if possible:

a): $2 (3 + a)$
b): $a (5 + b)$
c): $\frac{1}{3} \cdot (3 + b)$
d): $(2 + a) \cdot \frac{3}{4}$
e): $b (\frac{2}{3} + \frac{1}{3})$
f): $\frac{3}{8} (a + b)$
g): $(b - \frac{2 b}{9}) 6$
h): $\frac{8}{3} (\frac{a}{3} + b)$

1.7)

Expand the parentheses and contract the resulting expressions:

a): $(x + 2) (y - 2 x)$
b): $(2 x^{2} - a) (b - a)$
c): $(a + b) (a + b)$
d): $(a - b) (a - b)$
e): $(a - b) (a + b)$
f): $(x + y) (x + y)$
g): $(x + y) (x + y) (x + y)$
h): $2 (a + 3 b) (a + b)$
i): $5 (2 a + b) (3 a - 2 b)$
j): $3 (4 x + 6 y) (x + 2 y) (3 x + 5 y)$
k): $7 (2 x^{2} + 3 y) (2 x + 5 y)$
l): $2 (6 x^{2} + 2 y^{2}) (3 x^{2} + y^{2})$
m): $(a + b) (a + b) (a + b) (a + b)$

1.8)

Expand the parentheses and contract the resulting expressions:

a): $x (x + 2) + (x + 2) (x + 1)$
b): $(x + 4) (x + 5) - 3 x (5 + 2 x)$
c): $\begin{aligned} (3 x - 2) (5 + 2 x) \\ + (4 x + 4) (6 x + 1) \end{aligned}$
d): $\begin{aligned} (4 x + 2) (3 + 2 x) \\ - (9 x + 1) (2 x - 3) \end{aligned}$
e): $\begin{aligned} (5 + 4 x) (2 + 6 x) \\ - (x - 7) (3 x - 5) \end{aligned}$
f): $(5 + 4 x) (2 + 6 x) + (x - 7) (3 x - 5)$

a): $x (x + 2) + (x + 2) (x + 1)$
b): $(x + 4) (x + 5) - 3 x (5 + 2 x)$
c): $(3 x - 2) (5 + 2 x) + (4 x + 4) (6 x + 1)$
d): $(4 x + 2) (3 + 2 x) - (9 x + 1) (2 x - 3)$
e): $(5 + 4 x) (2 + 6 x) - (x - 7) (3 x - 5)$
f): $(5 + 4 x) (2 + 6 x) + (x - 7) (3 x - 5)$

1.9)

Expand the parentheses and contract the resulting expressions:

a): $\begin{aligned} 2 (a + b) (3 a + b) \\ + 3 (2 a + b) (a - b) \end{aligned}$
b): $\begin{aligned} 7 (4 a - 2 b) (2 a - b) \\ - (a - 4 b) (2 a - b) \end{aligned}$
c): $\begin{aligned} 4 (2 a - b) (3 a - 2 b) \\ - 3 (- 5 a - (- 3) \cdot b) (a + b) \end{aligned}$
d): $\begin{aligned} (a + b) (2 a - 3 b) \\ - 5 (- 2 a - 2 b) (6 a + b) \\ \cdot (3 a - 4 b) \end{aligned}$
e): $\begin{aligned} (2 a - 5) (3 b + 1) \\ - (4 a + 3 b - 2) (a + b) \\ + 5 (3 a + b) \end{aligned}$

a): $2 (a + b) (3 a + b) + 3 (2 a + b) (a - b)$
b): $7 (4 a - 2 b) (2 a - b) - (a - 4 b) (2 a - b)$
c): $4 (2 a - b) (3 a - 2 b) - 3 (- 5 a - (- 3) \cdot b) (a + b)$
d): $(a + b) (2 a - 3 b) - 5 (- 2 a - 2 b) (6 a + b) (3 a - 4 b)$
e): $(2 a - 5) (3 b + 1) - (4 a + 3 b - 2) (a + b) + 5 (3 a + b)$

1.10)

Expand the parentheses and contract the resulting expressions:

a): $2 x + (3 + 4 x)$
b): $x + 2 (x + 5)$
c): $7 x + 2 (6 x + 11 + 2 x) + x$
d): $5 + 3 x + 2 (x + 2) + 4 x$
e): $\begin{aligned} 7 - 2 x + 3 (2 x + 5) \\ - (- 2 + (- 7) + 3 x) \end{aligned}$
f): $3 x - 2 (x + 3)$
g): $4 x - 3 (2 - x)$
h): $x + 5 - 4 (- 1 + 3 x)$

a): $2 x + (3 + 4 x)$
b): $x + 2 (x + 5)$
c): $7 x + 2 (6 x + 11 + 2 x) + x$
d): $5 + 3 x + 2 (x + 2) + 4 x$
e): $7 - 2 x + 3 (2 x + 5) - (- 2 + (- 7) + 3 x)$
f): $3 x - 2 (x + 3)$
g): $4 x - 3 (2 - x)$
h): $x + 5 - 4 (- 1 + 3 x)$

1.11)

Expand the parentheses and contract the resulting expressions:

a): $5 x^{2} - 3 x (2 x - 3) (4 x + 2)$
b): $- 7 x + 3 - 5 x (x + 2) (2 x - 1)$
c): $x^{2} - x + 2 + 3 (2 - 3 x) (5 x + 3)$
d): $\begin{aligned} x^{2} - 7 x (2 x + 3) (2 x - 3) \\ - (5 x - 3) \end{aligned}$
e): $\begin{aligned} 2 x (4 x - 5) (6 x + 3) \\ - x (x + 4) (3 x - 1) \end{aligned}$
f): $\begin{aligned} (2 x - 3) (3 x + 5) \\ - 3 x (4 x - 2) (3 x - 2 \end{aligned}$

a): $5 x^{2} - 3 x (2 x - 3) (4 x + 2)$
b): $- 7 x + 3 - 5 x (x + 2) (2 x - 1)$
c): $x^{2} - x + 2 + 3 (2 - 3 x) (5 x + 3)$
d): $x^{2} - 7 x (2 x + 3) (2 x - 3) - (5 x - 3)$
e): $2 x (4 x - 5) (6 x + 3) - x (x + 4) (3 x - 1)$
f): $(2 x - 3) (3 x + 5) - 3 x (4 x - 2) (3 x - 2)$

1.12)

Simplify the expressions as much as possible:

a): $\frac{2 (4 x + 5 x)}{x (12 - 3)}$
b): $\frac{4 x (2 + 3)}{5 x - x (7 - 6)}$
c): $\frac{18 y}{(15 - 9) y}$
d): $\frac{- (9 x + 4 x)}{2 (20 x + 6 x)}$
e): $\frac{5 x (x + 2 x)}{(9 - 4) x}$
f): $\frac{(y + 5 y) 3 y}{4 (15 y - 8 y + 2 y)}$
g): $\frac{y (3 + 8) (- 4 + 12)}{5 (6 y - 3 y)}$
h): $\frac{8 (11 x - 9 x)}{4 - (12 - 16)}$

a): $\frac{2 (4 x + 5 x)}{x (12 - 3)}$
b): $\frac{4 x (2 + 3)}{5 x - x (7 - 6)}$
c): $\frac{18 y}{(15 - 9) y}$
d): $\frac{- (9 x + 4 x)}{2 (20 x + 6 x)}$
e): $\frac{5 x (x + 2 x)}{(9 - 4) x}$
f): $\frac{(y + 5 y) 3 y}{4 (15 y - 8 y + 2 y)}$
g): $\frac{y (3 + 8) (- 4 + 12)}{5 (6 y - 3 y)}$
h): $\frac{8 (11 x - 9 x)}{4 - (12 - 16)}$

1.13)

Contract the expressions:

a): $\begin{aligned} a^{2} + 4 (3 a + b) (- 7 a + 2 b) \\ + 3 b^{2} - (2 + 5 b) \end{aligned}$
b): $\begin{aligned} a b + 3 b - 4 + 2 (- 6 a - 3 b) \\ - 4 a b + (a + 2 b) (3 a - b) \\ + 5 \end{aligned}$
c): $\begin{aligned} 2 a^{2} + 3 \\ - (a + 2 b) (2 a - b) + 5 b^{2} \end{aligned}$
d): $\begin{aligned} 7 b^{2} + 5 - (a + b) (a + 2 b) \\ - (3 a + b) (2 a - 2 b) \\ + 7 (a + 1) \end{aligned}$

a): $a^{2} + 4 (3 a + b) (- 7 a + 2 b) + 3 b^{2} - (2 + 5 b)$
b): $a b + 3 b - 4 + 2 (- 6 a - 3 b) - 4 a b + (a + 2 b) (3 a - b) + 5$
c): $2 a^{2} + 3 - (a + 2 b) (2 a - b) + 5 b^{2}$
d): $7 b^{2} + 5 - (a + b) (a + 2 b) - (3 a + b) (2 a - 2 b) + 7 (a + 1)$

1.14)

Contract the expressions:

a): $\frac{a + 4 b}{2} + \frac{3 a - 7 b}{3}$
b): $\frac{3 x - 5 y}{3} + \frac{8 y + 4 x}{4}$
c): $\frac{6 y + 9 x}{5} - \frac{2 x - 3 y}{6}$
d): $\frac{12 b + 4 a}{8} + \frac{7 a - 5 b}{5}$
e): $\frac{14 b - 7 a}{9} - \frac{4 b - 3 a}{3}$
f): $\frac{20 m + 12 n}{14} + \frac{6 n - 8 m}{5}$
g): $\frac{2 (4 p + 3 a)}{8} - \frac{3 (p - 4 a)}{6}$
h): $- \frac{5 k + 5 h}{4} + \frac{17 k + 22 h}{12}$

a): $\frac{a + 4 b}{2} + \frac{3 a - 7 b}{3}$
b): $\frac{3 x - 5 y}{3} + \frac{8 y + 4 x}{4}$
c): $\frac{6 y + 9 x}{5} - \frac{2 x - 3 y}{6}$
d): $\frac{12 b + 4 a}{8} + \frac{7 a - 5 b}{5}$
e): $\frac{14 b - 7 a}{9} - \frac{4 b - 3 a}{3}$
f): $\frac{20 m + 12 n}{14} + \frac{6 n - 8 m}{5}$
g): $\frac{2 (4 p + 3 a)}{8} - \frac{3 (p - 4 a)}{6}$
h): $- \frac{5 k + 5 h}{4} + \frac{17 k + 22 h}{12}$

1.15)

Contract the expressions:

a): $\frac{2 (4 x + 3)}{a} + \frac{3 (7 - 2 x)}{b}$
b): $\frac{5 (x - 2)}{3 a} - \frac{(5 x + 3)}{3 b}$
c): $\frac{9 p + 3 d}{c} + \frac{6 (7 p - 4)}{5 c}$
d): $\frac{4 (12 x + 3 z)}{x} - \frac{10 (z + 3)}{4}$
e): $\frac{2 x (3 a - 6 b)}{t} + \frac{a (4 x - b)}{r}$
f): $\frac{3 p (2 + p)}{p^{2}} - \frac{3 (2 p - p)}{p}$
g): $\frac{(4 - x) (x + 3)}{4 x} + \frac{(0 - 3 x) 5}{8}$
h): $\frac{(3 a + 4 b) (5 b - 2 a)}{6 x} + \frac{(8 a + 2 b) (3 a - 4 b)}{4 x}$

1.16)

Expand the expressions:

a): $\frac{(4 x - 3 b)}{5 a} \cdot \frac{(b + 7 x)}{3 c}$
b): $\frac{(2 a + 5 b)}{12 d} \cdot \frac{(8 a + 3 b)}{3 k}$
c): $\frac{(6 b - 3 a)}{4 c} \cdot \frac{(9 a - 5 b)}{7 c}$
d): $\frac{(8 a + 2 b)}{7 n} \cdot \frac{(4 a - 6 b)}{3 n}$
e): $\frac{2 (2 a + 2 b)}{(a + b)} \cdot \frac{(2 a + 2 b)}{(a - b)}$
f): $\frac{4 (g + 5 h)}{(3 a - 5 b)} \cdot \frac{9 (2 h + 3 g)}{(2 b - 3 a)}$
g): $\frac{(15 n + 12 m)}{(6 b - 4 a)} \cdot \frac{(2 n - m)}{(2 a + 3 b)}$
h): $\frac{9 p}{(a - b)} \cdot \frac{(4 s - 6 r) 3}{(a - b)}$

a): $\frac{(4 x - 3 b)}{5 a} \cdot \frac{(b + 7 x)}{3 c}$
b): $\frac{(2 a + 5 b)}{12 d} \cdot \frac{(8 a + 3 b)}{3 k}$
c): $\frac{(6 b - 3 a)}{4 c} \cdot \frac{(9 a - 5 b)}{7 c}$
d): $\frac{(8 a + 2 b)}{7 n} \cdot \frac{(4 a - 6 b)}{3 n}$
e): $\frac{2 (2 a + 2 b)}{(a + b)} \cdot \frac{(2 a + 2 b)}{(a - b)}$
f): $\frac{4 (g + 5 h)}{(3 a - 5 b)} \cdot \frac{9 (2 h + 3 g)}{(2 b - 3 a)}$
g): $\frac{(15 n + 12 m)}{(6 b - 4 a)} \cdot \frac{(2 n - m)}{(2 a + 3 b)}$
h): $\frac{9 p}{(a - b)} \cdot \frac{(4 s - 6 r) 3}{(a - b)}$

You can find the solutions here.

Combining the Rules

The First and Second Algebraic Identity