1.1 Real numbers: algebra essentials (Page 3/35)

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Using the order of operations

Use the order of operations to evaluate each of the following expressions.

${(3 \cdot 2)}^{2} - 4 (6 + 2)$
$\frac{5^{2} - 4}{7} - \sqrt{11 - 2}$
$6 - | 5 - 8 | + 3 (4 - 1)$
$\frac{14 - 3 \cdot 2}{2 \cdot 5 - 3^{2}}$
$7 (5 \cdot 3) - 2 [(6 - 3) - 4^{2}] + 1$

$\begin{matrix} {(3 \cdot 2)}^{2} - 4 (6 + 2) & = & {(6)}^{2} - 4 (8) & Simplify parentheses \\ = & 36 - 4 (8) & Simplify exponent \\ = & 36 - 32 & Simplify multiplication \\ = & 4 & Simplify subtraction \end{matrix}$
$\begin{matrix} \frac{5^{2} \cdot 4}{7} - \sqrt{11 - 2} & = & \frac{5^{2} - 4}{7} - \sqrt{9} & Simplify grouping symbols (radical) \\ = & \frac{5^{2} - 4}{7} - 3 & Simplify radical \\ = & \frac{25 - 4}{7} - 3 & Simplify exponent \\ = & \frac{21}{7} - 3 & Simplify subtraction in numerator \\ = & 3 - 3 & Simplify division \\ = & 0 & Simplify subtraction \end{matrix}$
Note that in the first step, the radical is treated as a grouping symbol, like parentheses. Also, in the third step, the fraction bar is considered a grouping symbol so the numerator is considered to be grouped.
$\begin{matrix} 6 - | 5 - 8 | + 3 (4 - 1) & = & 6 - | −3 | + 3 (3) & Simplify inside grouping symbols \\ = & 6 - 3 + 3 (3) & Simplify absolute value \\ = & 6 - 3 + 9 & Simplify multiplication \\ = & 3 + 9 & Simplify subtraction \\ = & 12 & Simplify addition \end{matrix}$
$\begin{matrix} \frac{14 - 3 \cdot 2}{2 \cdot 5 - 3^{2}} & = & \frac{14 - 3 \cdot 2}{2 \cdot 5 - 9} & Simplify exponent \\ = & \frac{14 - 6}{10 - 9} & Simplify products \\ = & \frac{8}{1} & Simplify differences \\ = & 8 & Simplify quotient \end{matrix}$
In this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.
$\begin{matrix} 7 (5 \cdot 3) - 2 [(6 - 3) - 4^{2}] + 1 & = & 7 (15) - 2 [(3) - 4^{2}] + 1 & Simplify inside parentheses \\ = & 7 (15) - 2 (3 - 16) + 1 & Simplify exponent \\ = & 7 (15) - 2 (−13) + 1 & Subtract \\ = & 105 + 26 + 1 & Multiply \\ = & 132 & Add \end{matrix}$

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Try It

Use the order of operations to evaluate each of the following expressions.

$\sqrt{5^{2} - 4^{2}} + 7 {(5 - 4)}^{2}$
$1 + \frac{7 \cdot 5 - 8 \cdot 4}{9 - 6}$
$| 1.8 - 4.3 | + 0.4 \sqrt{15 + 10}$
$\frac{1}{2} [5 \cdot 3^{2} - 7^{2}] + \frac{1}{3} \cdot 9^{2}$
$[{(3 - 8)}^{2} - 4] - (3 - 8)$

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Using properties of real numbers

For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.

Commutative properties

The commutative property of addition states that numbers may be added in any order without affecting the sum.

a + b = b + a

We can better see this relationship when using real numbers.

\begin{array}{l} (−2) + 7 = 5 & and & 7 + (−2) = 5 \end{array}

Similarly, the commutative property of multiplication states that numbers may be multiplied in any order without affecting the product.

a \cdot b = b \cdot a

Again, consider an example with real numbers.

\begin{matrix} (−11) \cdot (−4) = 44 & and & (−4) \cdot (−11) = 44 \end{matrix}

It is important to note that neither subtraction nor division is commutative. For example, $17 - 5$ is not the same as $5 - 17.$ Similarly, $20 \div 5 \neq 5 \div 20.$

Associative properties

The associative property of multiplication tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.

a (b c) = (a b) c

Consider this example.

\begin{matrix} (3 \cdot 4) \cdot 5 = 60 & and & 3 \cdot (4 \cdot 5) = 60 \end{matrix}

The associative property of addition tells us that numbers may be grouped differently without affecting the sum.

a + (b + c) = (a + b) + c

This property can be especially helpful when dealing with negative integers. Consider this example.

\begin{matrix} [15 + (−9)] + 23 = 29 & and & 15 + [(−9) + 23] = 29 \end{matrix}

Are subtraction and division associative? Review these examples.

\begin{matrix} 8 - (3 - 15) & \overset{?}{=} & (8 - 3) - 15 & 64 \div (8 \div 4) & \overset{?}{=} & (64 \div 8) \div 4 \\ 8 - (- 12) & = & 5 - 15 & 64 \div 2 & \overset{?}{=} & 8 \div 4 \\ 20 & \neq & 20 - 10 & 32 & \neq & 2 \end{matrix}

Questions & Answers

how did you get 1640

Noor Reply

If auger is pair are the roots of equation x2+5x-3=0

Peter Reply

Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.

MATTHEW Reply

420

Sharon

from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph

George

Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28

Salma

Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.

Aaron Reply

find product (-6m+6) ( 3m²+4m-3)

SIMRAN Reply

-42m²+60m-18

Salma

what is the solution

bill

how did you arrive at this answer?

bill

-24m+3+3mÁ^2

Susan

i really want to learn

Amira

I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please

Amira

Hw did u arrive to this answer.

Aphelele

Bajemah

-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18

Salma

complete the table of valuesfor each given equatio then graph. 1.x+2y=3

Jovelyn Reply

x=3-2y

Salma

y=x+3/2

Salma

Enock

given that (7x-5):(2+4x)=8:7find the value of x

Nandala

3x-12y=18

Kelvin

please why isn't that the 0is in ten thousand place

Grace Reply

please why is it that the 0is in the place of ten thousand

Grace

Send the example to me here and let me see

Stephen

A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg

Marry Reply

how far

Abubakar

cool u

Enock

state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°

Abegail Reply

hello

BenJay

Method

I am eliacin, I need your help in maths

Rood

how can I help

Sir

hmm can we speak here?

Amoon

however, may I ask you some questions about Algarba?

Amoon

Enock

what the last part of the problem mean?

Roger

The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.

cameron Reply

Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.

mahnoor Reply

I'm guessing, but it's somewhere around $4335.00 I think

Lewis

12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00

Munster

difference between rational and irrational numbers

Arundhati Reply

When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?

Jakoiya Reply

how to reduced echelon form

Solomon Reply

Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?

Zack Reply

d=r×t the equation would be 8/r+24/r+4=3 worked out

Sheirtina

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Source: OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6

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