2.4: The Distributive property of Multiplication over Addition and/or Subtraction
- Page ID
- 13970
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The Distributive property of Multiplication over Addition and/or Subtraction
(0,8)(5,-5) (10,0)\(\overbrace{5\ \cdot\ (4}\ +\ 3)\ =(5\ \cdot \ 4\ )\ +\ (5\ \cdot \ 3 )\) (11.05,1.5)(0,1)1.5 (11.05,3.0)(1,0)5.5 (16.5,3.0)(0,-1)2.0 (10.1,-1.5)(1,0)3.2 (10.1,-0.5)(0,-1)1.0 (13.3,-0.5)(0,-1)1.0 (11.7,-1.5)(0,-1)1.0 (11.7,-2.5)(1,0)8.8 (20.4,-2.5)(0,1)2.2
\(5(4+3)=5(7)=35\)
and
\((5)(4)+(5)(3)=20+15=35\)
also.
The picture illustrates the Distributive property of multiplication over addition.
In general
\[\fbox{\Large \boldmath a(b+c)=ab+ac}\]
where \(a\), \(b\), and \(c\) are any real numbers.
A common mistake:
(0,4)(0,-5) (0,-2)\(x(y\pm z)=xy\pm z\) (0,-4)\(5(7+2)=5(7)+2\) (0,-6)\(5(9)=35+2\) (0,-8)\(45=37\) (0,-8)(1,1)7 (0,-1)(1,-1)7
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The picture illustrates the Distributive property of multiplication over Subtraction.
(0,8)(10,-4) (10,0)\(\overbrace{5\ \cdot\ (4\ }-\ 3)\ =(5\ \cdot \ 4\ )\ -\ (5\ \cdot \ 3 )\) (10.4,-1.4)(0,1)1 (10.4,-1.4)(1,0)4.7 (15.0,-1.4)(0,1)1 (25.6,-2.5)(0,1)2 (13.4,-2.4)(0,1)1 (13.4,-2.6)(1,0)12.3
(11.6,1.5)(0,1)1.1 (11.7,2.5)(1,0)7.6 (19.4,2.5)(0,-1)1.0
\(5(4-3)=5(1)=5\) and \((5)(4)-(5)(3)=20-15=5\) also.
In general
\[\fbox{\large \boldmath a(b+c)=ab+ac}\]
[-15pt]
where \(a\), \(b\), and \(c\) are any real numbers.
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Waning:
You cannot distribute multiplication over multiplication.
(0,8)(10,-4) (10,0)\(\overbrace{5\ \cdot\ (4\ }\cdot \ 3)\ =(5\ \cdot \ 4\ )\ \cdot\ (5\ \cdot \ 3 )\) (10.4,-1.4)(0,1)1 (10.4,-1.4)(1,0)4.3 (14.7,-1.4)(0,1)1 (24.4,-2.5)(0,1)2 (13.4,-2.4)(0,1)1 (13.4,-2.6)(1,0)11.1 (11.6,1.5)(0,1)1.1 (11.7,2.5)(1,0)7.6 (19.4,2.5)(0,-1)1.0 (10,-3)(5,2)15 (10,3)(5,-2)15
\(5(4\cdot3)=5(12)=60\) (Left side only)
and
\((5)(4)\cdot(5)(3)=20\cdot 15=300\) (Right side only)
The example shows that the distributive property does not apply to multiplication over multiplication.
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(0,8)(-10,-4) (9.8,0)\(\overbrace{16\ \div}\ (4\ \div\ 2)\ =(16\ \div \ 4\ )\ \div\ (16\ \div \ 2 )\) (14.2,-0.5)(0,-1)1 (14.2,-1.5)(1,0)8.7 (23.0,-1.5)(0,1)1 (10.9,1.5)(0,1)2 (10.9,2.5)(1,0)2.0 (12.9,2.5)(0,-1)1 (10.9,3.5)(1,0)4.5 (15.3,3.5)(0,-1)2.0
\(16\div (4\div 2)=16\div 2=8\)
and
\((16\div 4)\div(16\div 2)=4\div 8=\displaystyle \frac{1}{2}\)
which is different.
The example shows that the distributive property does not apply to division over division.
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