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Distributive Property Definition

In math, distributive property says that the sum of two or more addends multiplied past a number gives you the same answer equally distributing the multiplier, multiplying each addend separately, and calculation the products together.

PEMDAS vs. Distributive Belongings

PEMDAS Club of Operations	Using Distributive Property
$(5+7+3)\times 4$	$(\mathrm{v}+7+3)\times 4$
$=\mathrm{fifteen}\times 4$	$=(5\times 4)+(7\times \mathrm{iv})+(\mathrm{three}\times \mathrm{four})$
$=\mathrm{threescore}$	$=\mathrm{twenty}+78+12$
	$=60$

Table of Contents

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1. Definition
2. What Is Distributive Property
   • Distributive Holding of Multiplication
   • Distributive Holding of Segmentation
3. Examples
4. How To Use Distributive Property
   • Algebra
   • Geometry

What Is Distributive Property?

Distributive property is i of the almost used properties in mathematics. It is used to simplify and solve multiplication equations by distributing the multiplier to each number in the parentheses and then adding those products together to go your answer.

Distributive Property Steps

	Pace
$7\times \left(7+5+8\right)=?$	Given equation
$7\left(7\right)+7\left(5\right)+\mathrm{seven}\left(8\right)=?$	Distribute the multiplier
$49+35+56=?$	Multiply
$\mathbf{=}\mathbf{140}$	Add

Distributive property connects three basic mathematic operations in two pairings: multiplication and improver; and multiplication and subtraction.

The Distributive Holding states that, for real numbers $a$, $b$, and $c$, 2 conditions are always truthful:

• $a(b+c)=ab+ac$
• $a(b-c)=ab-ac$

You lot tin use distributive property to turn i complex multiplication equation into two simpler multiplication problems, and then add or subtract the two answers as required.

Distributive Holding of Multiplication

The distributive property is the aforementioned as the distributive property of multiplication, and it tin can be used over add-on or subtraction.

Here are examples of the distributive holding of multiplication at work:

Distributive Belongings of Addition and Subtraction

Distributive Property Over Addition	Distributive Property Over Subtraction
$6\times (10+\mathrm{v})$	$\mathrm{half\; dozen}\times (10-\mathrm{v})$
$=(6\times \mathrm{x})+(6\times 5)$	$=(6\times 10)-(\mathrm{vi}\times \mathrm{v})$
$=\mathrm{sixty}+30$	$=60-30$
$=90$	$=30$

Distributive Property of Partition

The distributive belongings does not employ to sectionalisation in the aforementioned since every bit it does with multiplication, but the idea of distributing or "breaking apart" tin can be used in partitioning.

The distributive law of sectionalisation can be used to simplify partitioning problems by breaking apart or distributing the numerator into smaller amounts to make the sectionalization issues easier to solve.

Instead of trying to solve $\frac{125}{5}$, you tin can use the distributive police force of partition to simplify the numerator and turn this 1 trouble into three smaller, easier division bug that you lot tin solve much easier.

$\frac{\mathrm{fifty}}{5}+\frac{50}{5}+\frac{25}{5}$

$10+\mathrm{ten}+5$

$\mathbf{25}$

Distributive Property Examples

These example problems that may help you to understand the power of the Distributive Belongings:

1. $\mathrm{eleven}\times (10+5)=?$
2. $11(10+\mathrm{v})=?$
3. $11\left(10\right)+11\left(\mathrm{five}\right)=?$
4. $110+55=?$

As well like shooting fish in a barrel? Let's endeavor a real-life discussion problem using coin amounts:

You buy nine boxed lunches for the members of Math Club at $\$\mathrm{7.ninety}$ each. Using mental math, how much should you be reimbursed for the lunches? You observe that $\$\mathrm{7.ninety}$ is only $\$0.10$ away from $\$8$, so you use the Distributive Property:

• $9\left(\$7.90\right)=?$
• $9(\$8-\$0.10)=?$
• $9\left(\$8\right)-\mathrm{nine}\left(\$0.10\right)=?$
• $\$72-\$0.90=\$71.10$

The Math Gild treasurer should reimburse you lot $\$\mathrm{71.ten}$ for the lunches.

How To Use Distributive Property?

In basic operations, the Distributive Property applies to multiplication of the multiplicand to all terms inside parentheses. This is truthful whether you add together or subtract terms:

$\mathrm{ii}(\mathrm{three}+\mathrm{iv}+\mathrm{v})-6(\mathrm{seven}+8)=?$

The Distributive Holding allows you lot to distribute the multiplicands or factors exterior the parentheses (in this case, $\mathrm{ii}$ and $-6$), to each term inside the parentheses:

$2\left(\mathrm{three}\right)+\mathrm{two}\left(\mathrm{iv}\right)+2\left(5\right)-6\left(7\right)-6\left(8\right)=?$

$\mathrm{half\; dozen}+8+10-42-48=?$

$24-90=-66$

You lot can use the characteristics of the Distributive Belongings to "interruption autonomously" something that is also hard to practise equally mental math, as well:

$9\times \mathrm{one,847}=?$

Aggrandize the multiplier and distribute the multiplicand to each place value:

$9\left(\mathrm{1,000}\right)+9\left(800\right)+9\left(40\right)+\mathrm{ix}\left(7\right)=?$

$\mathrm{9,000}+\mathrm{7,200}+360+63=?$

Associate (grouping) addends for easier mental addition:

$(\mathrm{9,000}+\mathrm{seven,200})+(360+63)=?$

$\mathrm{16,200}+423=\mathrm{xvi,623}$

Distributive Property Algebra

In algebra, the Distributive Property is used to aid you lot simplify algebraic expressions, combine like terms, and find the value of variables. This works with monomials and when multiplying ii binomials:

$3(5a+12)-(a+8)=?$

Distribute the $\mathrm{iii}$ and the $-1$:

$3\left(\mathrm{five}a\right)+\mathrm{iii}\left(12\right)+(-1)(a+\mathrm{eight})=?$

$15a+36-a-8=?$

Combine similar terms:

$14a+28=?$

Subtract $28$ from both sides:

$\mathrm{xiv}a=-28$

Separate both sides by $\mathrm{fourteen}$:

$a=-2$

Here is another more example of how to utilize the distributive holding to simplify an algebraic expression:

$(3x+4)(x-7)$

$(\mathrm{three}\mathrm{10}+\mathrm{iv})(x+-7)$

Apply the Distributive Property:

$\left(3\mathrm{ten}\right)\left(x\right)+\left(\mathrm{iii}x\right)(-\mathrm{vii})+\left(4\right)\left(x\right)+\left(\mathrm{four}\right)(-7)$

Simplify:

$3x^{\mathrm{two}}-21x+4\mathrm{10}-28$

Combine like terms:

$3x^2-17\mathrm{10}-28$

You may be familiar with the steps to solving binomials as the FOIL method:

• F irst terms of each binomial are multiplied
• O uter terms — the beginning term of the first binomial and the second term of the second binomial are multiplied
• I nner terms — the second term of the first binomial and the first term of the 2d binomial are multiplied
• L ast terms — the terminal terms of each binomial are multiplied

Negative and Positive Signs with the Distributive Property

Distributive Property works with all real numbers, which includes positive and negative integers. In algebra especially, you lot need to pay careful attention to negative signs in expressions.

Review the steps above we used in this problem, to a higher place:

$3(\mathrm{five}a+12)-(a+8)=?$

We added a $+$ sign after the starting time term and distributed a $-1$ to a and $\mathrm{eight}$, like this:

$3(5a+12)+(-1)(a+8)=?$

We then distributed the negative sign to both terms within the 2nd parentheses:

$\mathrm{fifteen}a+36-a-\mathrm{eight}=?$

We can show this distribution of the negative sign with two generic formulas, one for addition and ane for subtraction:

• $-(a+b)=-a-b$
• $-(a-b)=-a+b$

Distributive Property in Geometry

Nosotros tin can apply the Distributive Belongings to geometry when working with bug involving expanse of rectangles. Though algebra may seem unrelated to geometry, the two fields are strongly connected.

Suppose we are presented with a cartoon that lacks numbers just does bear witness a relationship.

We have no idea what the width and length are, simply we are told that the rectangle has an surface area of $65$ foursquare meters. How do we calculate width and length?

We know that expanse is width times length ($w\times \mathrm{fifty}$), which in this example is $\mathrm{ten}$ for the width and $\mathrm{ten}+8$ for the length, or $\left(x\right)(x+8)$.

Write down what we know:

$\mathrm{ten}(x+8)=65m^{\mathrm{ii}}$

Distribute $x$:

${\mathrm{10}}^2+\mathrm{viii}\mathrm{10}=65{\mathrm{thousand}}^{\mathrm{ii}}$

Catechumen it to a quadratic equation (subtract to make one side equal to $0$):

${\mathrm{ten}}^{\mathrm{ii}}+\mathrm{eight}\mathrm{ten}-65=0$

Factor the quadratic equation:

$(x-\mathrm{five})(x+13)=0$

$\mathrm{10}-\mathrm{five}=0and\mathrm{ten}+\mathrm{xiii}=0$

$x=\mathrm{five}a\mathrm{north}d\mathrm{10}=-13$

Nosotros cannot take a negative number for a width or length, then the correct reply is that $x$, the width, is equal to $\mathrm{v}$. This means the length, $x+8$, is equal to $13$. We tin can cheque our piece of work:

$\left(\mathrm{v}m\right)\left(8m\right)=65k^{\mathrm{ii}}$

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