Answer :
Alright, let's evaluate each expression as per the given values.
### 2. Evaluate each expression using [tex]\( p = -\frac{1}{2}, q = -\frac{3}{4}, r = \frac{4}{3} \)[/tex].
#### a) [tex]\( p \cdot q \)[/tex] and [tex]\( q^2 \cdot r \)[/tex]
1. [tex]\( p \cdot q = -\frac{1}{2} \cdot -\frac{3}{4} = 0.375 \)[/tex]
2. [tex]\( q^2 \cdot r = \left(-\frac{3}{4}\right)^2 \cdot \frac{4}{3} = 0.75 \)[/tex]
#### b) [tex]\( r \cdot q \)[/tex]
[tex]\( r \cdot q = \frac{4}{3} \cdot -\frac{3}{4} = -1.0 \)[/tex]
#### c) [tex]\( q \cdot r \cdot p \)[/tex]
[tex]\( q \cdot r \cdot p = -\frac{3}{4} \cdot \frac{4}{3} \cdot -\frac{1}{2} = 0.5 \)[/tex]
#### d) [tex]\( p^2 \cdot r \)[/tex]
[tex]\( p^2 \cdot r = \left(-\frac{1}{2}\right)^2 \cdot \frac{4}{3} = 0.3333333333333333 \)[/tex]
### 3. Simplify each of the following.
#### a) [tex]\( \left(\frac{2}{3}-\left(-\frac{1}{3}\right)\right) \div \left(\frac{-3}{4} - \left(-\frac{2}{3}\right)\right) \)[/tex]
[tex]\( = -11.999999999999995 \)[/tex]
#### b) [tex]\( \left(\frac{-1}{8}+\frac{1}{-4}\right) \left(\frac{-1}{-2}+\frac{1}{-6}\right) \)[/tex]
(first expression labeled as b)
[tex]\( = -0.125 \)[/tex]
#### c) [tex]\( \left(\frac{2}{-3} - \left(-\frac{1}{2}\right)\right) \div \left(\frac{-3}{2} - \frac{3}{2}\right) \)[/tex]
(second expression labeled as b)
[tex]\( = 0.055555555555555546 \)[/tex]
#### d) [tex]\( \left(\frac{4}{5}-\frac{3}{-4}\right) \left(\frac{-1}{3}-\frac{1}{4}\right) \)[/tex]
[tex]\( = -0.9041666666666666 \)[/tex]
### 4. Find the value of the expression [tex]\( x^2 - p \cdot q \)[/tex] for the following.
#### a) [tex]\( x = \frac{1}{2}, p = -\frac{1}{3}, q = -\frac{1}{4} \)[/tex]
[tex]\( x^2 - p \cdot q = \left(\frac{1}{2}\right)^2 - \left(-\frac{1}{3} \cdot -\frac{1}{4}\right) = 0.16666666666666669 \)[/tex]
#### b) [tex]\( x = -\frac{3}{4}, p = \frac{1}{2}, q = -\frac{2}{3} \)[/tex]
[tex]\( x^2 - p \cdot q = \left(-\frac{3}{4}\right)^2 - \left(\frac{1}{2} \cdot -\frac{2}{3}\right) = 0.8958333333333333 \)[/tex]
### 5. Evaluate each expression using [tex]\( x = -\frac{1}{2}, y = -\frac{3}{4}, z = \frac{4}{3} \)[/tex].
#### a) [tex]\( x \cdot y + z \)[/tex]
[tex]\( x \cdot y + z = -\frac{1}{2} \cdot -\frac{3}{4} + \frac{4}{3} = 1.7083333333333333 \)[/tex]
#### b) [tex]\( y - x \cdot z \)[/tex]
[tex]\( y - x \cdot z = -\frac{3}{4} - \left(-\frac{1}{2} \cdot \frac{4}{3}\right) = -0.08333333333333337 \)[/tex]
#### c) [tex]\( y \cdot z \div x \)[/tex]
[tex]\( y \cdot z \div x = \left(-\frac{3}{4} \cdot \frac{4}{3}\right) \div -\frac{1}{2} = 2.0 \)[/tex]
#### d) [tex]\( 2 \cdot x - 3 \cdot y \)[/tex]
[tex]\( 2 \cdot x - 3 \cdot y = 2 \cdot -\frac{1}{2} - 3 \cdot -\frac{3}{4} = 1.25 \)[/tex]
#### e) [tex]\( \frac{1}{2} \cdot y - \frac{2}{3} \cdot x \)[/tex]
[tex]\( \frac{1}{2} \cdot y - \frac{2}{3} \cdot x = \frac{1}{2} \cdot -\frac{3}{4} - \frac{2}{3} \cdot -\frac{1}{2} = -0.041666666666666685 \)[/tex]
#### f) [tex]\( \frac{3}{4} \cdot x - \frac{3}{8} \cdot y \)[/tex]
[tex]\( \frac{3}{4} \cdot x - \frac{3}{8} \cdot y = \frac{3}{4} \cdot -\frac{1}{2} - \frac{3}{8} \cdot -\frac{3}{4} = -0.09375 \)[/tex]
#### g) [tex]\( 2 \cdot x + 3 \cdot y - 2 \cdot z \)[/tex]
[tex]\( 2 \cdot x + 3 \cdot y - 2 \cdot z = 2 \cdot -\frac{1}{2} + 3 \cdot -\frac{3}{4} - 2 \cdot \frac{4}{3} = -5.916666666666666 \)[/tex]
#### h) [tex]\( \frac{1}{4} \cdot x - \frac{2}{3} \cdot y \div (4 \cdot z) \)[/tex]
[tex]\( \frac{1}{4} \cdot x - \frac{2}{3} \cdot y \div (4 \cdot z) = \frac{1}{4} \cdot -\frac{1}{2} - \frac{2}{3} \cdot -\frac{3}{4} \div (4 \cdot \frac{4}{3}) = -0.03125 \)[/tex]
These detailed steps should help you understand the process of evaluating each expression with the given values.
### 2. Evaluate each expression using [tex]\( p = -\frac{1}{2}, q = -\frac{3}{4}, r = \frac{4}{3} \)[/tex].
#### a) [tex]\( p \cdot q \)[/tex] and [tex]\( q^2 \cdot r \)[/tex]
1. [tex]\( p \cdot q = -\frac{1}{2} \cdot -\frac{3}{4} = 0.375 \)[/tex]
2. [tex]\( q^2 \cdot r = \left(-\frac{3}{4}\right)^2 \cdot \frac{4}{3} = 0.75 \)[/tex]
#### b) [tex]\( r \cdot q \)[/tex]
[tex]\( r \cdot q = \frac{4}{3} \cdot -\frac{3}{4} = -1.0 \)[/tex]
#### c) [tex]\( q \cdot r \cdot p \)[/tex]
[tex]\( q \cdot r \cdot p = -\frac{3}{4} \cdot \frac{4}{3} \cdot -\frac{1}{2} = 0.5 \)[/tex]
#### d) [tex]\( p^2 \cdot r \)[/tex]
[tex]\( p^2 \cdot r = \left(-\frac{1}{2}\right)^2 \cdot \frac{4}{3} = 0.3333333333333333 \)[/tex]
### 3. Simplify each of the following.
#### a) [tex]\( \left(\frac{2}{3}-\left(-\frac{1}{3}\right)\right) \div \left(\frac{-3}{4} - \left(-\frac{2}{3}\right)\right) \)[/tex]
[tex]\( = -11.999999999999995 \)[/tex]
#### b) [tex]\( \left(\frac{-1}{8}+\frac{1}{-4}\right) \left(\frac{-1}{-2}+\frac{1}{-6}\right) \)[/tex]
(first expression labeled as b)
[tex]\( = -0.125 \)[/tex]
#### c) [tex]\( \left(\frac{2}{-3} - \left(-\frac{1}{2}\right)\right) \div \left(\frac{-3}{2} - \frac{3}{2}\right) \)[/tex]
(second expression labeled as b)
[tex]\( = 0.055555555555555546 \)[/tex]
#### d) [tex]\( \left(\frac{4}{5}-\frac{3}{-4}\right) \left(\frac{-1}{3}-\frac{1}{4}\right) \)[/tex]
[tex]\( = -0.9041666666666666 \)[/tex]
### 4. Find the value of the expression [tex]\( x^2 - p \cdot q \)[/tex] for the following.
#### a) [tex]\( x = \frac{1}{2}, p = -\frac{1}{3}, q = -\frac{1}{4} \)[/tex]
[tex]\( x^2 - p \cdot q = \left(\frac{1}{2}\right)^2 - \left(-\frac{1}{3} \cdot -\frac{1}{4}\right) = 0.16666666666666669 \)[/tex]
#### b) [tex]\( x = -\frac{3}{4}, p = \frac{1}{2}, q = -\frac{2}{3} \)[/tex]
[tex]\( x^2 - p \cdot q = \left(-\frac{3}{4}\right)^2 - \left(\frac{1}{2} \cdot -\frac{2}{3}\right) = 0.8958333333333333 \)[/tex]
### 5. Evaluate each expression using [tex]\( x = -\frac{1}{2}, y = -\frac{3}{4}, z = \frac{4}{3} \)[/tex].
#### a) [tex]\( x \cdot y + z \)[/tex]
[tex]\( x \cdot y + z = -\frac{1}{2} \cdot -\frac{3}{4} + \frac{4}{3} = 1.7083333333333333 \)[/tex]
#### b) [tex]\( y - x \cdot z \)[/tex]
[tex]\( y - x \cdot z = -\frac{3}{4} - \left(-\frac{1}{2} \cdot \frac{4}{3}\right) = -0.08333333333333337 \)[/tex]
#### c) [tex]\( y \cdot z \div x \)[/tex]
[tex]\( y \cdot z \div x = \left(-\frac{3}{4} \cdot \frac{4}{3}\right) \div -\frac{1}{2} = 2.0 \)[/tex]
#### d) [tex]\( 2 \cdot x - 3 \cdot y \)[/tex]
[tex]\( 2 \cdot x - 3 \cdot y = 2 \cdot -\frac{1}{2} - 3 \cdot -\frac{3}{4} = 1.25 \)[/tex]
#### e) [tex]\( \frac{1}{2} \cdot y - \frac{2}{3} \cdot x \)[/tex]
[tex]\( \frac{1}{2} \cdot y - \frac{2}{3} \cdot x = \frac{1}{2} \cdot -\frac{3}{4} - \frac{2}{3} \cdot -\frac{1}{2} = -0.041666666666666685 \)[/tex]
#### f) [tex]\( \frac{3}{4} \cdot x - \frac{3}{8} \cdot y \)[/tex]
[tex]\( \frac{3}{4} \cdot x - \frac{3}{8} \cdot y = \frac{3}{4} \cdot -\frac{1}{2} - \frac{3}{8} \cdot -\frac{3}{4} = -0.09375 \)[/tex]
#### g) [tex]\( 2 \cdot x + 3 \cdot y - 2 \cdot z \)[/tex]
[tex]\( 2 \cdot x + 3 \cdot y - 2 \cdot z = 2 \cdot -\frac{1}{2} + 3 \cdot -\frac{3}{4} - 2 \cdot \frac{4}{3} = -5.916666666666666 \)[/tex]
#### h) [tex]\( \frac{1}{4} \cdot x - \frac{2}{3} \cdot y \div (4 \cdot z) \)[/tex]
[tex]\( \frac{1}{4} \cdot x - \frac{2}{3} \cdot y \div (4 \cdot z) = \frac{1}{4} \cdot -\frac{1}{2} - \frac{2}{3} \cdot -\frac{3}{4} \div (4 \cdot \frac{4}{3}) = -0.03125 \)[/tex]
These detailed steps should help you understand the process of evaluating each expression with the given values.
