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

Ayele, K., 2003. Introductory Economics, 3rd ed., Addis Ababa.

Widad Reply

can you send the book attached ?

Ariel

What is economics

Widad Reply

the study of how humans make choices under conditions of scarcity

AI-Robot

U(x,y) = (x×y)1/2 find mu of x for y

Desalegn Reply

U(x,y) = (xÃy)1/2 find mu of x for y

Desalegn

what is ecnomics

Jan Reply

this is the study of how the society manages it's scarce resources

Belonwu

what is macroeconomic

John Reply

macroeconomic is the branch of economics which studies actions, scale, activities and behaviour of the aggregate economy as a whole.

husaini

etc

husaini

difference between firm and industry

husaini Reply

what's the difference between a firm and an industry

Abdul

firm is the unit which transform inputs to output where as industry contain combination of firms with similar production 😅😅

Abdulraufu

Suppose the demand function that a firm faces shifted from Qd  120 3P to Qd  90  3P and the supply function has shifted from QS  20  2P to QS 10  2P . a) Find the effect of this change on price and quantity. b) Which of the changes in demand and supply is higher?

Toofiq Reply

explain standard reason why economic is a science

innocent Reply

factors influencing supply

Petrus Reply

what is economic.

Milan Reply

scares means__________________ends resources. unlimited

Jan

economics is a science that studies human behaviour as a relationship b/w ends and scares means which have alternative uses

Jan

calculate the profit maximizing for demand and supply

Zarshad Reply

Why qualify 28 supplies

Milan

what are explicit costs

Nomsa Reply

out-of-pocket costs for a firm, for example, payments for wages and salaries, rent, or materials

AI-Robot

concepts of supply in microeconomics

David Reply

economic overview notes

Amahle Reply

identify a demand and a supply curve

Salome Reply

i don't know

Parul

there's a difference

Aryan

Demand curve shows that how supply and others conditions affect on demand of a particular thing and what percent demand increase whith increase of supply of goods

Israr

Hi Sir please how do u calculate Cross elastic demand and income elastic demand?

Abari

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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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