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Multiply and Divide Integers

Multiply integers

Multiply Integers

Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.

We remember that $a\cdot b$ means add a, b times.

The next two examples are more interesting.

What does it mean to multiply 5 by $-3?$ It means subtract 5, 3 times. Looking at subtraction as “taking away,” it means to take away 5, 3 times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away 5 three times.

In summary:

\[\begin{array}{llllllll}5\cdot 3 & = & 15 & & & -5(3) & = & -15 \\ 5(-3) & = & -15 & & & (-5)(-3) & = & 15\end{array}\]

Notice that for multiplication of two signed numbers, when the:

signs are the same, the product is positive.
signs are different, the product is negative.

We’ll put this all together in the chart below.

Example

Try it.

Multiply: ⓐ $-9\cdot 3$ ⓑ $-2(-5)$ ⓒ $4(-8)$ ⓓ $7\cdot 6.$

Solution

ⓐ Multiply, noting that the signs are different so the product is negative.	$\begin{array}{l}-9\cdot 3 \\ -27\end{array}$
ⓑ Multiply, noting that the signs are the same so the product is positive.	$\begin{array}{l}-2(-5) \\ 10\end{array}$
ⓒ Multiply, with different signs.	$\begin{array}{l}4(-8) \\ -32\end{array}$
ⓓ Multiply, with same signs.	$\begin{array}{l}7\cdot 6 \\ 42\end{array}$

\[\begin{array}{lllllll} & & & -1\cdot 4 & & & -1(-3) \\ \text{Multiply.} & & & -4 & & & 3 \\ & & & -4\ \text{is the opposite of}\ 4. & & & 3\ \text{is the opposite of}\ -3.\end{array}\]

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Divide Integers

What about division? Division is the inverse operation of multiplication. So, $15\div 3=5$ because $5\cdot 3=15.$ In words, this expression says that 15 can be divided into three groups of five each because adding five three times gives 15. Look at some examples of multiplying integers, to figure out the rules for dividing integers.

\[\begin{array}{llllllllllllll}5\cdot 3 & = & 15\ \text{so}\ 15\div 3 & = & 5 & & & & & -5(3) & = & -15\ \text{so}\ -15\div 3 & = & -5 \\ (-5)(-3) & = & 15\ \text{so}\ 15\div (-3) & = & -5 & & & & & 5(-3) & = & -15\ \text{so}\ -15\div (-3) & = & 5\end{array}\]

Division follows the same rules as multiplication!

For division of two signed numbers, when the:

signs are the same, the quotient is positive.
signs are different, the quotient is negative.

And remember that we can always check the answer of a division problem by multiplying.

Example

Try it.

Divide: ⓐ $-27\div 3$ ⓑ $-100\div (-4).$

Solution

ⓐ Divide. With different signs, the quotient is negative.	$\begin{array}{l}-27\div 3 \\ -9\end{array}$
ⓑ Divide. With signs that are the same, the quotient is positive.	$\begin{array}{l}-100\div (-4) \\ 25\end{array}$

Simplify Expressions with Integers

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.

Example

Try it.

Simplify: $7(-2)+4(-7)-6.$

Solution

	$7(-2)+4(-7)-6$
Multiply first.	$-14+(-28)-6$
Add.	$-42-6$
Subtract.	$-48$

Example

Try it.

Simplify: ⓐ ${(-2)}^{4}$ ⓑ $\text{-}{2}^{4}.$

Solution

ⓐ Write in expanded form. Multiply. Multiply. Multiply.	$\begin{array}{l}{(-2)}^{4} \\ (-2)(-2)(-2)(-2) \\ 4(-2)(-2) \\ -8(-2) \\ 16\end{array}$
ⓑ Write in expanded form. We are asked to find the opposite of$\ {2}^{4}.$ Multiply. Multiply. Multiply.	$\begin{array}{l}\text{-}{2}^{4} \\ \text{-}(2\cdot 2\cdot 2\cdot 2) \\ \text{-}(4\cdot 2\cdot 2) \\ \text{-}(8\cdot 2) \\ -16\end{array}$

Notice the difference in parts ⓐ and ⓑ. In part ⓐ , the exponent means to raise what is in the parentheses, the $(-2)$ 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.

The next example reminds us to simplify inside parentheses first.

Example

Try it.

Simplify: $12-3(9-12).$

Solution

	$12-3(9-12)$
Subtract in parentheses first.	$12-3(-3)$
Multiply.	$12-(-9)$
Subtract.	$21$

Example

Try it.

Simplify: $8(-9)\div {(-2)}^{3}.$

Solution

	$8(-9)\div {(-2)}^{3}$
Exponents first.	$8(-9)\div (-8)$
Multiply.	$-72\div (-8)$
Divide.	$9$

Example

Try it.

Simplify: $-30\div 2+(-3)(-7).$

Solution

	$-30\div 2+(-3)(-7)$
Multiply and divide left to right, so divide first.	$-15+(-3)(-7)$
Multiply.	$-15+21$
Add.	$6$

Evaluate Variable Expressions with Integers

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.

Example

Try it.

When $n=-5,$ evaluate: ⓐ $n+1$ ⓑ $\text{-}n+1.$

Solution

ⓐ



Simplify.	−4

ⓑ



Simplify.
Add.	6

Example

Try it.

Evaluate ${(x+y)}^{2}$ when $x=-18$ and $y=24.$

Solution



Add inside parenthesis.	(6)²
Simplify.	36

Example

Try it.

Evaluate $20-z$ when ⓐ $z=12$ and ⓑ $z=-12.$

Solution

ⓐ



Subtract.	8

ⓑ



Subtract.	32

Example

Try it.

Evaluate: $2{x}^{2}+3x+8$ when $x=4.$

Solution

Substitute $4\ \text{for}\ x.$ Use parentheses to show multiplication.


Substitute.
Evaluate exponents.
Multiply.
Add.	52

Translate Phrases to Expressions with Integers

Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.

Example

Try it.

Translate and simplify: the sum of 8 and $-12,$ increased by 3.

Solution

	the sum of 8 and $-12,$ increased by 3.
Translate.	$[8+(-12)]+3$
Simplify. Be careful not to confuse the brackets with an absolute value sign.	$(-4)+3$
Add.	$-1$

When we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.

$a-b$

$a$ minus $b$
the difference of $a$ and $b$
$b$ subtracted from $a$
$b$ less than $a$

Be careful to get a and b in the right order!

Example

Try it.

Translate and then simplify ⓐ the difference of 13 and $-21$ ⓑ subtract 24 from $-19.$

Solution

ⓐ Translate. Simplify.	$\begin{array}{l}\text{the}\ \text{difference}\ \text{of}\ 13\ \text{and}\ -21 \\ 13-(-21) \\ 34\end{array}$
ⓑ Translate. Remember, "subtract $b$ from $a$ means $a-b$. Simplify.	$\begin{array}{l}\text{subtract}\ 24\ \text{from}\ -19 \\ -19-24 \\ -43\end{array}$

Once again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is “product” and for division is “quotient.”

Example

Try it.

Translate to an algebraic expression and simplify if possible: the product of $-2$ and 14.

Solution

	$\text{the product}\ \text{of}\ -2\ \text{and}\ 14$
Translate.	$(-2)(14)$
Simplify.	$-28$

Example

Try it.

Translate to an algebraic expression and simplify if possible: the quotient of $-56$ and $-7.$

Solution

	$\text{the quotient}\ \text{of}\ -56\ \text{and}\ -7$
Translate.	$-56\div (-7)$
Simplify.	$8$

Use Integers in Applications

We’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

How to Apply a Strategy to Solve Applications with Integers

Try it.

In the morning, the temperature in Urbana, Illinois was 11 degrees. By mid-afternoon, the temperature had dropped to $-9$ degrees. What was the difference of the morning and afternoon temperatures?

Solution

Example

Try it.

The Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?

Solution

Step 1. Read the problem. Make sure all the words and ideas are understood.
Step 2. Identify what we are asked to find.	the number of yards lost
Step 3. Write a phrase that gives the information to find it.	three times a 15-yard penalty
Step 4. Translate the phrase to an expression.	$3(-15)$
Step 5. Simplify the expression.	$-45$
Step 6. Answer the question with a complete sentence.	The team lost 45 yards.

Key Concepts

Multiplication and Division of Two Signed Numbers
- Same signs—Product is positive
- Different signs—Product is negative
Strategy for Applications
1. Identify what you are asked to find.
2. Write a phrase that gives the information to find it.
3. Translate the phrase to an expression.
4. Simplify the expression.
5. Answer the question with a complete sentence.

Multiply and Divide Integers

Multiply Integers

In the following exercises, multiply.

Try it.

$-4\cdot 8$

Solution

$-32$

Try it.

$-3\cdot 9$

Try it.

$9(-7)$

Solution

$-63$

Try it.

$13(-5)$

Try it.

$-1⋅6$

Solution

$-6$

Try it.

$-1⋅3$

Try it.

$-1(-14)$

Solution

Try it.

$-1(-19)$

Divide Integers

In the following exercises, divide.

Try it.

$-24\div 6$

Solution

$-4$

Try it.

$35\div (-7)$

Try it.

$-52\div (-4)$

Solution

Try it.

$-84\div (-6)$

Try it.

$-180\div 15$

Solution

$-12$

Try it.

$-192\div 12$

Simplify Expressions with Integers

In the following exercises, simplify each expression.

Try it.

$5(-6)+7(-2)-3$

Solution

$-47$

Try it.

$8(-4)+5(-4)-6$

Try it.

${(-2)}^{6}$

Solution

Try it.

${(-3)}^{5}$

Try it.

$\text{-}{4}^{2}$

Solution

$-16$

Try it.

$\text{-}{6}^{2}$

Try it.

$-3(-5)(6)$

Solution

Try it.

$-4(-6)(3)$

Try it.

$(8-11)(9-12)$

Solution

Try it.

$(6-11)(8-13)$

Try it.

$26-3(2-7)$

Solution

Try it.

$23-2(4-6)$

Try it.

$65\div (-5)+(-28)\div (-7)$

Solution

$-9$

Try it.

$52\div (-4)+(-32)\div (-8)$

Try it.

$9-2[3-8(-2)]$

Solution

$-29$

Try it.

$11-3[7-4(-2)]$

Try it.

${(-3)}^{2}-24\div (8-2)$

Solution

Try it.

${(-4)}^{2}-32\div (12-4)$

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

Try it.

$y+(-14)$ when ⓐ $y=-33$ ⓑ $y=30$

Solution

ⓐ $-47$ ⓑ 16

Try it.

$x+(-21)$ whenⓐ $x=-27$ⓑ $x=44$

Try it.

ⓐ $a+3$ when $a=-7$
ⓑ $\text{-}a+3$ when $a=-7$

Solution

ⓐ $-4$ ⓑ 10

Try it.

ⓐ $d+(-9)$ when $d=-8$
ⓑ $\text{-}d+(-9)$ when $d=-8$

Try it.

$m+n$ when
$m=-15,n=7$

Solution

$-8$

Try it.

$p+q$ when
$p=-9,q=17$

Try it.

$r+s$ when $r=-9,s=-7$

Solution

$-16$

Try it.

$t+u$ when $t=-6,u=-5$

Try it.

${(x+y)}^{2}$ when
$x=-3,y=14$

Solution

121

Try it.

${(y+z)}^{2}$ when
$y=-3,z=15$

Try it.

$-2x+17$ when

ⓐ $x=8$
ⓑ $x=-8$

Solution

ⓐ 1
ⓑ 33

Try it.

$-5y+14$ when
ⓐ $y=9$
ⓑ $y=-9$

Try it.

$10-3m$ when
ⓐ $m=5$
ⓑ $m=-5$

Solution

ⓐ $-5$ ⓑ 25

Try it.

$18-4n$ when
ⓐ $n=3$
ⓑ $n=-3$

Try it.

$2{w}^{2}-3w+7$ when
$w=-2$

Solution

Try it.

$3{u}^{2}-4u+5$ when $u=-3$

Try it.

$9a-2b-8$ when
$a=-6\ \text{and}\ b=-3$

Solution

$-56$

Try it.

$7m-4n-2$ when
$m=-4\ \text{and}\ n=-9$

Translate English Phrases to Algebraic Expressions

In the following exercises, translate to an algebraic expression and simplify if possible.

Try it.

the sum of 3 and $-15,$ increased by 7

Solution

$(3+(-15))+7;-5$

Try it.

the sum of $-8$ and $-9,$ increased by 23

Try it.

the difference of 10 and $-18$

Solution

$10-(-18);28$

Try it.

subtract 11 from $-25$

Try it.

the difference of $-5$ and $-30$

Solution

$-5-(-30);25$

Try it.

subtract $-6$ from $-13$

Try it.

the product of $\text{-3 and 15}$

Solution

$-3\cdot 15;-45$

Try it.

the product of $\text{-4 and 16}\$

Try it.

the quotient of $-60$ and $-20$

Solution

$-60\div (-20);3$

Try it.

the quotient of $-40$ and $-20$

Try it.

the quotient of $-6$ and the sum of a and b

Solution

$\frac{-6}{a+b}$

Try it.

the quotient of $-7$ and the sum of m and n

Try it.

the product of $-10$ and the difference of $p\ \text{and}\ q$

Solution

$-10(p-q)$

Try it.

the product of $-13$ and the difference of $c\ \text{and}\ d$

Use Integers in Applications

In the following exercises, solve.

Try it.

Temperature On January $15,$ the high temperature in Anaheim, California, was $84\text{^{\circ}}.$ That same day, the high temperature in Embarrass, Minnesota was $-12\text{^{\circ}}.$ What was the difference between the temperature in Anaheim and the temperature in Embarrass?

Solution

$96\text{^{\circ}}$

Try it.

Temperature On January $21,$ the high temperature in Palm Springs, California, was $89\text{^{\circ}},$ and the high temperature in Whitefield, New Hampshire was $-31\text{^{\circ}}.$ What was the difference between the temperature in Palm Springs and the temperature in Whitefield?

Try it.

Football On the first down, the Chargers had the ball on their 25-yard line. They lost 6 yards on the first-down play, gained 10 yards on the second-down play, and lost 8 yards on the third-down play. What was the yard line at the end of the third-down play?

Solution

Try it.

Football On first down, the Steelers had the ball on their 30-yard line. They gained 9 yards on the first-down play, lost 14 yards on the second-down play, and lost 2 yards on the third-down play. What was the yard line at the end of the third-down play?

Try it.

Checking Account Mayra has $124 in her checking account. She writes a check for $152. What is the new balance in her checking account?

Solution

$\text{-}\$28$

Try it.

Checking Account Selina has $165 in her checking account. She writes a check for $207. What is the new balance in her checking account?

Try it.

Checking Account Diontre has a balance of $\text{-}\$38$ in his checking account. He deposits $225 to the account. What is the new balance?

Solution

$187

Try it.

Checking Account Reymonte has a balance of $\text{-}\$49$ in his checking account. He deposits $281 to the account. What is the new balance?

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

उत्तर उघडा

ⓐ Multiply, noting that the signs are different so the product is negative.	$\begin{array}{l}-9\cdot 3 \\ -27\end{array}$
ⓑ Multiply, noting that the signs are the same so the product is positive.	$\begin{array}{l}-2(-5) \\ 10\end{array}$
ⓒ Multiply, with different signs.	$\begin{array}{l}4(-8) \\ -32\end{array}$
ⓓ Multiply, with same signs.	$\begin{array}{l}7\cdot 6 \\ 42\end{array}$

Multiply: ⓐ $-1\cdot 7$ ⓑ $-1(-11).$

उत्तर उघडा

ⓐ Multiply, noting that the signs are different so the product is negative.	$\begin{array}{l}-1\cdot 7 \\ -7 \\ -7\ \text{is the opposite of}\ 7.\end{array}$
ⓑ Multiply, noting that the signs are the same so the product is positive.	$\begin{array}{l}-1(-11) \\ 11 \\ 11\ \text{is the opposite of}\ -11.\end{array}$

Multiply: ⓐ $-1\cdot 9$ ⓑ $-1\cdot (-17).$

उत्तर उघडा

ⓐ $-9$ ⓑ 17
Multiply: ⓐ $-1\cdot 8$ ⓑ $-1\cdot (-16).$

उत्तर उघडा

ⓐ $-8$ ⓑ 16

Divide: ⓐ $-27\div 3$ ⓑ $-100\div (-4).$

उत्तर उघडा

ⓐ Divide. With different signs, the quotient is negative.	$\begin{array}{l}-27\div 3 \\ -9\end{array}$
ⓑ Divide. With signs that are the same, the quotient is positive.	$\begin{array}{l}-100\div (-4) \\ 25\end{array}$

Divide: ⓐ $-42\div 6$ ⓑ $-117\div (-3).$

उत्तर उघडा

ⓐ $-7$ ⓑ 39
Divide: ⓐ $-63\div 7$ ⓑ $-115\div (-5).$

उत्तर उघडा

ⓐ $-9$ ⓑ 23
Simplify: $7(-2)+4(-7)-6.$

उत्तर उघडा

$7(-2)+4(-7)-6$
Multiply first. $-14+(-28)-6$
Add. $-42-6$
Subtract. $-48$
Simplify: $8(-3)+5(-7)-4.$

उत्तर उघडा

$-63$
Simplify: $9(-3)+7(-8)-1.$

उत्तर उघडा

$-84$

Simplify: ⓐ ${(-2)}^{4}$ ⓑ $\text{-}{2}^{4}.$

उत्तर उघडा

ⓐ Write in expanded form. Multiply. Multiply. Multiply.	$\begin{array}{l}{(-2)}^{4} \\ (-2)(-2)(-2)(-2) \\ 4(-2)(-2) \\ -8(-2) \\ 16\end{array}$
ⓑ Write in expanded form. We are asked to find the opposite of$\ {2}^{4}.$ Multiply. Multiply. Multiply.	$\begin{array}{l}\text{-}{2}^{4} \\ \text{-}(2\cdot 2\cdot 2\cdot 2) \\ \text{-}(4\cdot 2\cdot 2) \\ \text{-}(8\cdot 2) \\ -16\end{array}$

Simplify: ⓐ ${(-3)}^{4}$ ⓑ $\text{-}{3}^{4}.$

उत्तर उघडा

ⓐ 81 ⓑ $-81$
Simplify: ⓐ ${(-7)}^{2}$ ⓑ $\text{-}{7}^{2}.$

उत्तर उघडा

ⓐ 49 ⓑ $-49$
Simplify: $12-3(9-12).$

उत्तर उघडा

$12-3(9-12)$
Subtract in parentheses first. $12-3(-3)$
Multiply. $12-(-9)$
Subtract. $21$
Simplify: $17-4(8-11).$

उत्तर उघडा

29
Simplify: $16-6(7-13).$

उत्तर उघडा

52
Simplify: $8(-9)\div {(-2)}^{3}.$

उत्तर उघडा

$8(-9)\div {(-2)}^{3}$
Exponents first. $8(-9)\div (-8)$
Multiply. $-72\div (-8)$
Divide. $9$
Simplify: $12(-9)\div {(-3)}^{3}.$

उत्तर उघडा

4
Simplify: $18(-4)\div {(-2)}^{3}.$

उत्तर उघडा

9
Simplify: $-30\div 2+(-3)(-7).$

उत्तर उघडा

$-30\div 2+(-3)(-7)$
Multiply and divide left to right, so divide first. $-15+(-3)(-7)$
Multiply. $-15+21$
Add. $6$
Simplify: $-27\div 3+(-5)(-6).$

उत्तर उघडा

21
Simplify: $-32\div 4+(-2)(-7).$

उत्तर उघडा

6
When $n=-5,$ evaluate: ⓐ $n+1$ ⓑ $\text{-}n+1.$

उत्तर उघडा

ⓐ

Simplify. −4
ⓑ

Simplify.
Add. 6
When $n=-8,$ evaluate ⓐ $n+2$ ⓑ $\text{-}n+2.$

उत्तर उघडा

ⓐ $-6$ ⓑ 10
When $y=-9,$ evaluate ⓐ $y+8$ ⓑ $\text{-}y+8.$

उत्तर उघडा

ⓐ $-1$ ⓑ 17
Evaluate ${(x+y)}^{2}$ when $x=-18$ and $y=24.$

उत्तर उघडा

Add inside parenthesis. (6)²
Simplify. 36
Evaluate ${(x+y)}^{2}$ when $x=-15$ and $y=29.$

उत्तर उघडा

196
Evaluate ${(x+y)}^{3}$ when $x=-8$ and $y=10.$

उत्तर उघडा

8
Evaluate $20-z$ when ⓐ $z=12$ and ⓑ $z=-12.$

उत्तर उघडा

ⓐ
Subtract. 8

ⓑ
Subtract. 32
Evaluate: $17-k$ when ⓐ $k=19$ and ⓑ $k=-19.$

उत्तर उघडा

ⓐ $-2$ ⓑ 36
Evaluate: $-5-b$ when ⓐ $b=14$ and ⓑ $b=-14.$

उत्तर उघडा

ⓐ $-19$ ⓑ 9
Evaluate: $2{x}^{2}+3x+8$ when $x=4.$

उत्तर उघडा

Substitute $4\ \text{for}\ x.$ Use parentheses to show multiplication.

Substitute.
Evaluate exponents.
Multiply.
Add. 52
Evaluate: $3{x}^{2}-2x+6$ when $x=-3.$

उत्तर उघडा

39
Evaluate: $4{x}^{2}-x-5$ when $x=-2.$

उत्तर उघडा

13
Translate and simplify: the sum of 8 and $-12,$ increased by 3.

उत्तर उघडा

the sum of 8 and $-12,$ increased by 3.
Translate. $[8+(-12)]+3$
Simplify. Be careful not to confuse the brackets with an absolute value sign. $(-4)+3$
Add. $-1$
Translate and simplify the sum of 9 and $-16,$ increased by 4.

उत्तर उघडा

$(9+(-16))+4;-3$
Translate and simplify the sum of $-8$ and $-12,$ increased by 7.

उत्तर उघडा

$(-8+(-12))+7;-13$

Translate and then simplify ⓐ the difference of 13 and $-21$ ⓑ subtract 24 from $-19.$

उत्तर उघडा

ⓐ Translate. Simplify.	$\begin{array}{l}\text{the}\ \text{difference}\ \text{of}\ 13\ \text{and}\ -21 \\ 13-(-21) \\ 34\end{array}$
ⓑ Translate. Remember, "subtract $b$ from $a$ means $a-b$. Simplify.	$\begin{array}{l}\text{subtract}\ 24\ \text{from}\ -19 \\ -19-24 \\ -43\end{array}$

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Multiply and Divide Integers

Multiply integers
Divide integers
Simplify expressions with integers
Evaluate variable expressions with integers
Translate English phrases to algebraic expressions
Use integers in applications
signs are the
signs are

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

स्वतःचा प्रयत्न करा

Parts of this page are adapted from OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

अधिक माहिती Algebra

Linear equations Quadratic equations Systems of equations Inequalities Factoring Expanding Simplifying expressions Functions and graphs Exponential and logarithmic equations Polynomial equations Absolute value

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.

	\(7(-2)+4(-7)-6\)
Multiply first.	\(-14+(-28)-6\)
Add.	\(-42-6\)
Subtract.	\(-48\)

ⓐ Write in expanded form. Multiply. Multiply. Multiply.	\(\begin{array}{l}{(-2)}^{4} \\ (-2)(-2)(-2)(-2) \\ 4(-2)(-2) \\ -8(-2) \\ 16\end{array}\)
ⓑ Write in expanded form. We are asked to find the opposite of\(\ {2}^{4}.\) Multiply. Multiply. Multiply.	\(\begin{array}{l}\text{-}{2}^{4} \\ \text{-}(2\cdot 2\cdot 2\cdot 2) \\ \text{-}(4\cdot 2\cdot 2) \\ \text{-}(8\cdot 2) \\ -16\end{array}\)

	\(12-3(9-12)\)
Subtract in parentheses first.	\(12-3(-3)\)
Multiply.	\(12-(-9)\)
Subtract.	\(21\)

Same signs	Product	Example
Two positives Two negatives	Positive Positive	\(\begin{array}{lll}7\cdot 4 & = & 28 \\ -8(-6) & = & 48\end{array}\)

ⓐ Multiply, noting that the signs are different so the product is negative.	\(\begin{array}{l}-9\cdot 3 \\ -27\end{array}\)
ⓑ Multiply, noting that the signs are the same so the product is positive.	\(\begin{array}{l}-2(-5) \\ 10\end{array}\)
ⓒ Multiply, with different signs.	\(\begin{array}{l}4(-8) \\ -32\end{array}\)
ⓓ Multiply, with same signs.	\(\begin{array}{l}7\cdot 6 \\ 42\end{array}\)

ⓐ Divide. With different signs, the quotient is negative.	\(\begin{array}{l}-27\div 3 \\ -9\end{array}\)
ⓑ Divide. With signs that are the same, the quotient is positive.	\(\begin{array}{l}-100\div (-4) \\ 25\end{array}\)

	\(8(-9)\div {(-2)}^{3}\)
Exponents first.	\(8(-9)\div (-8)\)
Multiply.	\(-72\div (-8)\)
Divide.	\(9\)

	\(-30\div 2+(-3)(-7)\)
Multiply and divide left to right, so divide first.	\(-15+(-3)(-7)\)
Multiply.	\(-15+21\)
Add.	\(6\)

	the sum of 8 and \(-12,\) increased by 3.
Translate.	\([8+(-12)]+3\)
Simplify. Be careful not to confuse the brackets with an absolute value sign.	\((-4)+3\)
Add.	\(-1\)

ⓐ Translate. Simplify.	\(\begin{array}{l}\text{the}\ \text{difference}\ \text{of}\ 13\ \text{and}\ -21 \\ 13-(-21) \\ 34\end{array}\)
ⓑ Translate. Remember, "subtract \(b\) from \(a\) means \(a-b\). Simplify.	\(\begin{array}{l}\text{subtract}\ 24\ \text{from}\ -19 \\ -19-24 \\ -43\end{array}\)

	\(\text{the product}\ \text{of}\ -2\ \text{and}\ 14\)
Translate.	\((-2)(14)\)
Simplify.	\(-28\)

	\(\text{the quotient}\ \text{of}\ -56\ \text{and}\ -7\)
Translate.	\(-56\div (-7)\)
Simplify.	\(8\)

ⓐ Multiply, noting that the signs are different so the product is negative.	\(\begin{array}{l}-1\cdot 7 \\ -7 \\ -7\ \text{is the opposite of}\ 7.\end{array}\)
ⓑ Multiply, noting that the signs are the same so the product is positive.	\(\begin{array}{l}-1(-11) \\ 11 \\ 11\ \text{is the opposite of}\ -11.\end{array}\)