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$\begin{array}{cc}(9y)4=9(y4)& \text{Both}\text{\hspace{0.17em}}\text{represent}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{same}\text{\hspace{0.17em}}\text{product}\text{.}\end{array}$
Fill in the $(\begin{array}{cc}& \end{array})$ to make each statement true. Use the associative properties.
$\left[(7m-2)(m+3)\right](m+4)=(7m-2)\left[(\begin{array}{cc}& \end{array})(\begin{array}{cc}& \end{array})\right]$
$(m+3)(m+4)$
Simplify (rearrange into a simpler form): $5x6b8ac4$ .
According to the commutative property of multiplication, we can make a series of consecutive switches and get all the numbers together and all the letters together.
$\begin{array}{ll}5\cdot 6\cdot 8\cdot 4\cdot x\cdot b\cdot a\cdot c\hfill & \hfill \\ 960xbac\hfill & \text{Multiply}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{numbers}\text{.}\hfill \\ 960abcx\hfill & \text{By}\text{\hspace{0.17em}}\text{convention,}\text{\hspace{0.17em}}\text{we}\text{\hspace{0.17em}}\text{will,}\text{\hspace{0.17em}}\text{when}\text{\hspace{0.17em}}\text{possible,}\text{\hspace{0.17em}}\text{write}\text{\hspace{0.17em}}\text{all}\text{\hspace{0.17em}}\text{letters}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}\text{alphabetical}\text{\hspace{0.17em}}\text{order}\text{.}\hfill \end{array}$
Simplify each of the following quantities.
When we were first introduced to multiplication we saw that it was developed as a description for repeated addition.
$4+4+4=3\cdot 4$
Notice that there are three 4’s, that is, 4 appears 3
times . Hence, 3 times 4.
We know that algebra is generalized arithmetic. We can now make an important generalization.
When a number
$a$ is added repeatedly
$n$ times, we have
$\underset{a\text{\hspace{0.17em}}\text{appears}\text{\hspace{0.17em}}n\text{\hspace{0.17em}}\text{times}}{\underbrace{a+a+a+\cdots +a}}$
Then, using multiplication as a description for repeated addition, we can replace
$\begin{array}{ccc}\underset{n\text{\hspace{0.17em}}\text{times}}{\underbrace{a+a+a+\cdots +a}}& \text{with}& na\end{array}$
For example:
$x+x+x+x$ can be written as $4x$ since $x$ is repeatedly added 4 times.
$x+x+x+x=4x$
$r+r$ can be written as $2r$ since $r$ is repeatedly added 2 times.
$r+r=2r$
The distributive property involves both multiplication and addition. Let’s rewrite $4(a+b).$ We proceed by reading $4(a+b)$ as a multiplication: 4 times the quantity $(a+b)$ . This directs us to write
$\begin{array}{lll}4(a+b)\hfill & =\hfill & (a+b)+(a+b)+(a+b)+(a+b)\hfill \\ \hfill & =\hfill & a+b+a+b+a+b+a+b\hfill \end{array}$
Now we use the commutative property of addition to collect all the $a\text{'}s$ together and all the $b\text{'}s$ together.
$\begin{array}{lll}4(a+b)\hfill & =\hfill & \underset{4a\text{'}s}{\underbrace{a+a+a+a}}+\underset{4b\text{'}s}{\underbrace{b+b+b+b}}\hfill \end{array}$
Now, using multiplication as a description for repeated addition, we have
$\begin{array}{lll}4(a+b)\hfill & =\hfill & 4a+4b\hfill \end{array}$
We have distributed the 4 over the sum to both $a$ and $b$ .
The distributive property is useful when we cannot or do not wish to perform operations inside parentheses.
Use the distributive property to rewrite each of the following quantities.
What property of real numbers justifies
$a(b+c)=(b+c)a?$
the commutative property of multiplication
Use the distributive property to rewrite each of the following quantities.
We summarize the identity properties as follows.
$\begin{array}{cc}\begin{array}{l}\text{ADDITIVE}\text{\hspace{0.17em}}\text{IDENTITY}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{PROPERTY}\end{array}& \begin{array}{l}\text{MULTIPLICATIVE}\text{\hspace{0.17em}}\text{IDENTITY}\\ \text{}\text{}\text{}\text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{PROPERTY}\end{array}\\ \text{If}\text{\hspace{0.17em}}a\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{a}\text{\hspace{0.17em}}\text{real}\text{\hspace{0.17em}}\text{number,\hspace{0.17em}then}& \text{If}\text{\hspace{0.17em}}a\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{a}\text{\hspace{0.17em}}\text{real}\text{\hspace{0.17em}}\text{number,}\text{\hspace{0.17em}}\text{then}\\ a+0=a\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0+a=a& a\cdot 1=a\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\cdot a=a\end{array}$
We summarize the inverse properties as follows.
Use the commutative property of addition and multiplication to write expressions for an equal number for the following problems. You need not perform any calculations.
$(8+a)(x+6)$
$(x+y)(x-y)$
$m(a+2b)$
$(21c)(0.008)$
$(5)(b-6)$
$\square \text{\hspace{0.17em}}\cdot \u25cb$
$\u25cb\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\square $
Simplify using the commutative property of multiplication for the following problems. You need not use the distributive property.
$5x10y5z$
$6d4e1f2(g+2h)$
$\left(\frac{1}{2}\right)d\left(\frac{1}{4}\right)e\left(\frac{1}{2}\right)a$
$\frac{1}{16}ade$
$3(a+6)2(a-9)6b$
$1(x+2y)(6+z)9(3x+5y)$
$9\left(x+2y\right)\left(6+z\right)\left(3x+5y\right)$
For the following problems, use the distributive property to expand the quantities.
$x(2y+5)$
$(1+\text{\hspace{0.17em}}d)e$
$c(2a+\text{\hspace{0.17em}}10b)$
$8y(12a+b)$
$(a+6)(x+y)$
$0.48(0.34a+0.61)$
( [link] ) Find the value of $4\cdot 2+5(2\cdot 4-6\xf73)-2\cdot 5$ .
( [link] ) Is the statement $3(5\cdot 3-3\cdot 5)+6\cdot 2-3\cdot 4<0$ true or false?
false
( [link] ) Draw a number line that extends from $-2$ to 2 and place points at all integers between and including $-2$ and 3.
( [link] ) Replace the $\ast $ with the appropriate relation symbol $(<,>).-7\ast -3$ .
$<$
( [link] ) What whole numbers can replace $x$ so that the statement $-2\le x<2$ is true?
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