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Division follows the same rules as multiplication!
For division of two signed numbers, when the:
And remember that we can always check the answer of a division problem by multiplying.
For multiplication and division of two signed numbers:
Same signs | Result |
---|---|
Two positives
Two negatives |
Positive
Positive |
If the signs are the same, the result is positive. |
Different signs | Result |
---|---|
Positive and negative
Negative and positive |
Negative
Negative |
If the signs are different, the result is negative. |
Divide: ⓐ $\mathrm{-27}\xf73$ ⓑ $\mathrm{-100}\xf7\left(\mathrm{-4}\right).$
Divide: ⓐ $\mathrm{-42}\xf76$ ⓑ $\mathrm{-117}\xf7\left(\mathrm{-3}\right).$
ⓐ $\mathrm{-7}$ ⓑ 39
Divide: ⓐ $\mathrm{-63}\xf77$ ⓑ $\mathrm{-115}\xf7\left(\mathrm{-5}\right).$
ⓐ $\mathrm{-9}$ ⓑ 23
What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?
Let’s try some examples. We’ll simplify expressions that use all four operations with integers—addition, subtraction, multiplication, and division. Remember to follow the order of operations.
Simplify: $7\left(\mathrm{-2}\right)+4\left(\mathrm{-7}\right)-6.$
$\begin{array}{cccccc}& & & & & \hfill 7\left(\mathrm{-2}\right)+4\left(\mathrm{-7}\right)-6\hfill \\ \text{Multiply first.}\hfill & & & & & \hfill \mathrm{-14}+\left(\mathrm{-28}\right)-6\hfill \\ \text{Add.}\hfill & & & & & \hfill \mathrm{-42}-6\hfill \\ \text{Subtract.}\hfill & & & & & \hfill \mathrm{-48}\hfill \end{array}$
Simplify: $8\left(\mathrm{-3}\right)+5\left(\mathrm{-7}\right)-4.$
$\mathrm{-63}$
Simplify: $9\left(\mathrm{-3}\right)+7\left(\mathrm{-8}\right)-1.$
$\mathrm{-84}$
Simplify: ⓐ ${\left(\mathrm{-2}\right)}^{4}$ ⓑ $\text{\u2212}{2}^{4}.$
Notice the difference in parts ⓐ and ⓑ . In part ⓐ , the exponent means to raise what is in the parentheses, the $\left(\mathrm{-2}\right)$ to the ${4}^{\text{th}}$ power. In part ⓑ , the exponent means to raise just the 2 to the ${4}^{\text{th}}$ power and then take the opposite.
Simplify: ⓐ ${\left(\mathrm{-3}\right)}^{4}$ ⓑ $\text{\u2212}{3}^{4}.$
ⓐ 81 ⓑ $\mathrm{-81}$
Simplify: ⓐ ${\left(\mathrm{-7}\right)}^{2}$ ⓑ $\text{\u2212}{7}^{2}.$
ⓐ 49 ⓑ $\mathrm{-49}$
The next example reminds us to simplify inside parentheses first.
Simplify: $12-3\left(9-12\right).$
$\begin{array}{cccccc}& & & & & \hfill 12-3\left(9-12\right)\hfill \\ \text{Subtract in parentheses first.}\hfill & & & & & \hfill 12-3\left(\mathrm{-3}\right)\hfill \\ \text{Multiply.}\hfill & & & & & \hfill 12-\left(\mathrm{-9}\right)\hfill \\ \text{Subtract.}\hfill & & & & & \hfill 21\hfill \end{array}$
Simplify: $8\left(\mathrm{-9}\right)\xf7{\left(\mathrm{-2}\right)}^{3}.$
$\begin{array}{cccccc}& & & & & \hfill 8\left(\mathrm{-9}\right)\xf7{\left(\mathrm{-2}\right)}^{3}\hfill \\ \text{Exponents first.}\hfill & & & & & \hfill 8\left(\mathrm{-9}\right)\xf7\left(\mathrm{-8}\right)\hfill \\ \text{Multiply.}\hfill & & & & & \hfill \mathrm{-72}\xf7\left(\mathrm{-8}\right)\hfill \\ \text{Divide.}\hfill & & & & & \hfill 9\hfill \end{array}$
Simplify: $12\left(\mathrm{-9}\right)\xf7{\left(\mathrm{-3}\right)}^{3}.$
4
Simplify: $18\left(\mathrm{-4}\right)\xf7{\left(\mathrm{-2}\right)}^{3}.$
9
Simplify: $\mathrm{-30}\xf72+\left(\mathrm{-3}\right)\left(\mathrm{-7}\right).$
$\begin{array}{cccccc}& & & & & \hfill \mathrm{-30}\xf72+\left(\mathrm{-3}\right)\left(\mathrm{-7}\right)\hfill \\ \text{Multiply and divide left to right, so divide first.}\hfill & & & & & \hfill \mathrm{-15}+\left(\mathrm{-3}\right)\left(\mathrm{-7}\right)\hfill \\ \text{Multiply.}\hfill & & & & & \hfill \mathrm{-15}+21\hfill \\ \text{Add.}\hfill & & & & & \hfill 6\hfill \end{array}$
Simplify: $\mathrm{-27}\xf73+\left(\mathrm{-5}\right)\left(\mathrm{-6}\right).$
21
Simplify: $\mathrm{-32}\xf74+\left(\mathrm{-2}\right)\left(\mathrm{-7}\right).$
6
Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.
When $n=\mathrm{-5},$ evaluate: ⓐ $n+1$ ⓑ $\text{\u2212}n+1.$
ⓐ
Simplify. | −4 |
ⓑ
Simplify. | |
Add. | 6 |
When $n=\mathrm{-8},$ evaluate ⓐ $n+2$ ⓑ $\text{\u2212}n+2.$
ⓐ $\mathrm{-6}$ ⓑ 10
When $y=\mathrm{-9},$ evaluate ⓐ $y+8$ ⓑ $\text{\u2212}y+8.$
ⓐ $\mathrm{-1}$ ⓑ 17
Evaluate ${\left(x+y\right)}^{2}$ when $x=\mathrm{-18}$ and $y=24.$
Add inside parenthesis. | (6) ^{2} |
Simplify. | 36 |
Evaluate ${\left(x+y\right)}^{2}$ when $x=\mathrm{-15}$ and $y=29.$
196
Evaluate ${\left(x+y\right)}^{3}$ when $x=\mathrm{-8}$ and $y=10.$
8
Evaluate $20-z$ when ⓐ $z=12$ and ⓑ $z=\mathrm{-12}.$
ⓐ
Subtract. | 8 |
Subtract. | 32 |
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