Properties of Real Numbers
Learning Outcomes
- Use the order of operations to simplify an algebraic expression.
- Use properties of real numbers to simplify algebraic expressions.
How was the fraction above simplified?
Recall, when mulitplying fractions, we multiply the numerators together and place the result over the product of the denominators.- [latex]\dfrac{a}{b}\cdot\dfrac{c}{d}=\dfrac{ac}{bd}[/latex]
- [latex]6\cdot\dfrac{2}{3}=\dfrac{6}{1}\cdot\dfrac{2}{3}=\dfrac{12}{3}=4[/latex]
A General Note: Order of Operations
Operations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the acronym PEMDAS: P(arentheses) E(xponents) M(ultiplication) and D(ivision) A(ddition) and S(ubtraction)How To: Given a mathematical expression, simplify it using the order of operations.
- Simplify any expressions within grouping symbols.
- Simplify any expressions containing exponents or radicals.
- Perform any multiplication and division in order, from left to right.
- Perform any addition and subtraction in order, from left to right.
Example: Using the Order of Operations
Use the order of operations to evaluate each of the following expressions.- [latex]{\left(3\cdot 2\right)}^{2}-4\left(6+2\right)[/latex]
- [latex]\dfrac{{5}^{2}-4}{7}-\sqrt{11 - 2}[/latex]
- [latex]6-|5 - 8|+3\left(4 - 1\right)[/latex]
- [latex]\dfrac{14 - 3\cdot 2}{2\cdot 5-{3}^{2}}[/latex]
- [latex]7\left(5\cdot 3\right)-2\left[\left(6 - 3\right)-{4}^{2}\right]+1[/latex]
Answer: 1.
[latex]\begin{align}\left(3\cdot 2\right)^{2} & =\left(6\right)^{2}-4\left(8\right) && \text{Simplify parentheses} \\ & =36-4\left(8\right) && \text{Simplify exponent} \\ & =36-32 && \text{Simplify multiplication} \\ & =4 && \text{Simplify subtraction}\end{align}[/latex]
2.[latex]\begin{align}\frac{5^{2}-4}{7}-\sqrt{11-2} & =\frac{5^{2}-4}{7}-\sqrt{9} && \text{Simplify grouping systems (radical)} \\ & =\frac{5^{2}-4}{7}-3 && \text{Simplify radical} \\ & =\frac{25-4}{7}-3 && \text{Simplify exponent} \\ & =\frac{21}{7}-3 && \text{Simplify subtraction in numerator} \\ & =3-3 && \text{Simplify division} \\ & =0 && \text{Simplify subtraction}\end{align}[/latex]
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. 3.[latex]\begin{align}6-|5-8|+3\left(4-1\right) & =6-|-3|+3\left(3\right) && \text{Simplify inside grouping system} \\ & =6-3+3\left(3\right) && \text{Simplify absolute value} \\ & =6-3+9 && \text{Simplify multiplication} \\ & =3+9 && \text{Simplify subtraction} \\ & =12 && \text{Simplify addition}\end{align}[/latex]
4.[latex]\begin{align}\frac{14-3\cdot2}{2\cdot5-3^{2}} & =\frac{14-3\cdot2}{2\cdot5-9} && \text{Simplify exponent} \\ & =\frac{14-6}{10-9} && \text{Simplify products} \\ & =\frac{8}{1} && \text{Simplify quotient} \\ & =8 && \text{Simplify quotient}\end{align}[/latex] In this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.
5.[latex]\begin{align}7\left(5\cdot3\right)-2[\left(6-3\right)-4^{2}]+1 & =7\left(15\right)-2[\left(3\right)-4^{2}]+1 && \text{Simplify inside parentheses} \\ & 7\left(15\right)-2\left(3-16\right)+1 && \text{Simplify exponent} \\ & =7\left(15\right)-2\left(-13\right)+1 && \text{Subtract} \\ & =105+26+1 && \text{Multiply} \\ & =132 && \text{Add}\end{align}[/latex]
Try It
Use the order of operations to evaluate the following expression: [latex-display]\sqrt{(21-5)}-2(7+9^{2})+\dfrac{23-5}{3^2}[/latex-display] Watch this short video to learn how to evaluate a mathematical expression with an online graphing calculator. https://youtu.be/uM13cNyQGPM Check your work with an online graphing calculator. [embed] [embed]Recall: simplifying fractions
To simplify a fraction, look for common factors in the numerator and the denominator. In the video below, a fraction, [latex]\dfrac{14}{26}[/latex] must be simplified. Since [latex]\dfrac{14}{26}=\dfrac{2\cdot7}{2\cdot13}[/latex], and since [latex]\dfrac{2}{2} = 1[/latex], this fraction simplifies to [latex]\dfrac{7}{13}[/latex].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.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.Distributive Property
The distributive property states that the product of a factor times a sum is the sum of the factor times each term in the sum.Identity Properties
The identity property of addition states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number.Inverse Properties
The inverse property of addition states that, for every real number a, there is a unique number, called the additive inverse (or opposite), denoted−a, that, when added to the original number, results in the additive identity, 0.A General Note: Properties of Real Numbers
The following properties hold for real numbers a, b, and c.Addition | Multiplication | |
---|---|---|
Commutative Property | [latex]a+b=b+a[/latex] | [latex]a\cdot b=b\cdot a[/latex] |
Associative Property | [latex]a+\left(b+c\right)=\left(a+b\right)+c[/latex] | [latex]a\left(bc\right)=\left(ab\right)c[/latex] |
Distributive Property | [latex]a\cdot \left(b+c\right)=a\cdot b+a\cdot c[/latex] | |
Identity Property | There exists a unique real number called the additive identity, 0, such that, for any real number a
[latex]a+0=a[/latex] |
There exists a unique real number called the multiplicative identity, 1, such that, for any real number a
[latex]a\cdot 1=a[/latex] |
Inverse Property | Every real number a has an additive inverse, or opposite, denoted –a, such that
[latex]a+\left(-a\right)=0[/latex] |
Every nonzero real number a has a multiplicative inverse, or reciprocal, denoted [latex]\frac{1}{a}[/latex], such that
[latex]a\cdot \left(\dfrac{1}{a}\right)=1[/latex] |
Using the Inverse property of numbers: simplifying fractions
Recall when simplifying fractions, we look for common factors in the numerator and denominator to "cancel out." What we mean by that is that common factors, [latex]\dfrac{a}{a}[/latex], divide to the number 1. This is based on the inverse property of multiplication. Because the inverse property states that [latex]a\cdot \left(\dfrac{1}{a}\right)=1[/latex], and because fraction multiplication gives that [latex]\dfrac{a}{b}\cdot\dfrac{c}{d} = \dfrac {ac}{bd}[/latex], We have [latex-display]\dfrac{a}{1}\cdot \left(\dfrac{1}{a}\right)=\dfrac{a}{a}=1[/latex-display]Example: Using Properties of Real Numbers
Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.- [latex]3\cdot 6+3\cdot 4[/latex]
- [latex]\left(5+8\right)+\left(-8\right)[/latex]
- [latex]6-\left(15+9\right)[/latex]
- [latex]\dfrac{4}{7}\cdot \left(\frac{2}{3}\cdot \dfrac{7}{4}\right)[/latex]
- [latex]100\cdot \left[0.75+\left(-2.38\right)\right][/latex]
Answer: 1. [latex-display]\begin{align}3\cdot6+3\cdot4 &=3\cdot\left(6+4\right) && \text{Distributive property} \\ &=3\cdot10 && \text{Simplify} \\ & =30 && \text{Simplify}\end{align}[/latex-display] 2. [latex-display]\begin{align}\left(5+8\right)+\left(-8\right) &=5+\left[8+\left(-8\right)\right] &&\text{Associative property of addition} \\ &=5+0 && \text{Inverse property of addition} \\ &=5 &&\text{Identity property of addition}\end{align}[/latex-display] 3. [latex-display]\begin{align}6-\left(15+9\right) & =6+(-15-9) && \text{Distributive property} \\ & =6+\left(-24\right) && \text{Simplify} \\ & =-18 && \text{Simplify}\end{align}[/latex-display] 4. [latex-display]\begin{align}\frac{4}{7}\cdot\left(\frac{2}{3}\cdot\frac{7}{4}\right) & =\frac{4}{7} \cdot\left(\frac{7}{4}\cdot\frac{2}{3}\right) && \text{Commutative property of multiplication} \\ & =\left(\frac{4}{7}\cdot\frac{7}{4}\right)\cdot\frac{2}{3} && \text{Associative property of multiplication} \\ & =1\cdot\frac{2}{3} && \text{Inverse property of multiplication} \\ & =\frac{2}{3} && \text{Identity property of multiplication}\end{align}[/latex-display] 5. [latex-display]\begin{align}100\cdot[0.75+\left(-2.38\right)] & =100\cdot0.75+100\cdot\left(-2.38\right) && \text{Distributive property} \\ & =75+\left(-238\right) && \text{Simplify} \\ & =-163 && \text{Simplify}\end{align}[/latex-display]
Try It
Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.- [latex]\left(-\dfrac{23}{5}\right)\cdot \left[11\cdot \left(-\dfrac{5}{23}\right)\right][/latex]
- [latex]5\cdot \left(6.2+0.4\right)[/latex]
- [latex]18-\left(7 - 15\right)[/latex]
- [latex]\dfrac{17}{18}+\cdot \left[\dfrac{4}{9}+\left(-\dfrac{17}{18}\right)\right][/latex]
- [latex]6\cdot \left(-3\right)+6\cdot 3[/latex]
Answer:
- 11, commutative property of multiplication, associative property of multiplication, inverse property of multiplication, identity property of multiplication;
- 33, distributive property;
- 26, distributive property;
- [latex]\dfrac{4}{9}[/latex], commutative property of addition, associative property of addition, inverse property of addition, identity property of addition;
- 0, distributive property, inverse property of addition, identity property of addition
Licenses & Attributions
CC licensed content, Original
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
- Evaluate a Mathematical Expression With the Desmos Calculator. Authored by: Lumen Learning. License: Public Domain: No Known Copyright.
CC licensed content, Shared previously
- Question ID 259. Authored by: Sousa, James. License: Other. License terms: IMathAS Community License CC-BY + GPL.
- Question ID 99379. Authored by: Davis, Desiree. License: Other. License terms: IMathAS Community License CC-BY + GPL.
- Question ID 92360. Authored by: Jenck, Michael for Lumen Learning. License: Other. License terms: IMathAS Community License CC-BY + GPL.
- Question ID 92361. Authored by: Jenck, Michael for Lumen Learning. License: Other. License terms: IMathAS Community License CC-BY + GPL.
- College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].
- Simplifying Expressions With Square Roots. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
CC licensed content, Specific attribution
- College Algebra. Provided by: OpenStax Authored by: OpenStax College Algebra. Located at: https://cnx.org/contents/[email protected]:1/Preface. License: CC BY: Attribution.