What Is the Distributive Property

What Is the Distributive Property SAT / ACT Prep Online Guides and Tips What is the distributive property? Did you go over the distributive property definition in school but still aren’t sure what it is or why it’s important? The distributive property is a key mathematical property you’ll need to know to solve many algebra problems. In this guide, we explain exactly what the distributive property is, why it’s important, when you should use it, what other math rules you need to know for it, and we also work through several examples so you can see the distributive property in action. What Is the Distributive Property? The distributive property, sometimes known as the distributive property of multiplication, tells us how to solve certain algebraic expressions that include both multiplication and addition. The literal definition of the distributive property is that multiplying a number by a sum is the same as doing each multiplication separately. In equation form, the distributive property looks like this: $a(b+c) = ab + ac$ (Remember, in math, when two numbers/factors are right next to each other, that means to multiply them.) Like many math definitions, the distributive property is easier to understand when you look at a few examples. Here’s a simple one: $$5 (2 +7)$$ Normally, if you had a problem like this, you’d add 2 and 7 together to get 9, then you’d multiply 5times 9 to get 45. This is the simplest way to solve the equation, and it also follows the order of operations, which tells you to simplify whatever is in the parentheses first before moving onto other operations like multiplication. Solving that equation using the distributive property would look like this: $$5 (2+7)$$ The distributive property means doing multiplication before the addition within the parentheses, so we’d distribute the 5 to both values within the parentheses: $$5(2) + 5(7)$$ Work out the multiplication: $$10 + 35$$ Then add the two numbers together: $$10+35=45$$ We get the same answer as we did solving the problem with the first method, which shows that the distributive property works. Now, why would you want to use the distributive property when it took longer than the first method? The distributive property comes in handy when you have terms within the parentheses that can’t be added together, such as this equation: ${3/4}(a + 2b)$. Because there are variables involved, there’s no easy way to simplify $a + 2b$. In these more complicated equations, the distributive property can help us get the equation into a form that makes it easier to simplify or solve. We’ll see examples of how to do this later on in this guide. 3 Key Rules Related to the Distributive Property When you’re using the distributive property, you’ll often have to use or be aware of other mathematical rules and properties in order to solve or simplify the equations. Here are three of the most important ones to know. Commutative Laws The commutative laws state that you can swap numbers when adding or multiplying and still get the same answer. So $x + y = y + x$ and $x(y) = y(x)$ These are likely intuitive for you by now, but they’re an important part of the distributive property, which wouldn’t work without them. You can use them when you need help simplifying certain equations in order to get them into a more workable form. Order of Operations When you have a complicated equation that looks like it can be simplified in multiple ways, the order of operations gives you the correct way to work through those operations. The acronym PEMDAS makes it easy to remember which operations to work on first. From first to last, here’s the order you should work out operations: Parentheses Exponents Multiplication and Division (do these at the same time, working left to right) Addition and Subtraction (do these at the same time, working left to right) The order of operations is important to know because you’ll often have to remember it when simplifying equations that include a lot of different operations. It can also help you determine whether to use the distributive property or not. Order of operations states the first step you should take when simplifying an equation is to work out whatever is in a parentheses set, but if what’s in the parentheses can’t be simplified, that’s a sign to use the distributive property. Quadratic Formula The quadratic formula states that, for $ax^2+ bx + c = 0$, the values of $x$ which are the solutions to the equation are given by: $$x={-b ±Ã¢Ë†Å¡{b^2-4ac}}/{2a}$$ When using the distributive property, you may be able to simplify some equations into the $ax^2 + bx + c = 0$ form so that you can use the quadratic equation to solve for $\bi x$. Distributive Property of Multiplication Example Problems In this section we go over three examples of simplifying problems using the distributive property. You’ll notice each of them contain variables in the parentheses, which is a key sign that the distributive property is needed. Example 1 $$\bo4\bi x(\bo5\bi x + \bo6) = -\bo7$$ First, we’re going to distribute $4x$ to both $5x$ and 6. $$4x(5x) + 4x(6) = -7$$ Now, multiply those out: $$20x^2+ 24x = 7$$ Add 7 to both sides: $$20x^2+ 24x +7 = 0$$ This equation is now in the proper formula to solve for $x$ using the quadratic formula (x would equal $-0.7$ and $-0.5$), or you might be able to keep the equation in that form if you were just being asked to simplify it. Example 2 $$\bo3\bi x(\bi x-\bo4) + \bo5(\bo4\bi x + \bo6)$$ For this equation, there are two sets of parentheses, so we need to use the distributive property twice. Distribute the 3x to its set of parentheses and the 5x to its set of parentheses: $$3x(x) + 3x(-4) + 5(4x) + 5(6)$$ Multiply it out: $$3x^2- 12x + 20x^2+ 30$$ Add the two $x^2$ terms together to simplify $$23x^2- 12x + 30$$ Example 3 $$-\bo7(\bi x + \bo4) + \bo8(\bo2 - \bo4\bi x)$$ This example is a bit trickier because the 7 has a negative sign in front of it. When the value just outside the parentheses is negative, the negative sign must be distributed to each term within the parentheses. Distribute the -7 to its set of parentheses and the 8 to its set of parentheses: $$(-7)(x) + (-7)(4) + (8)(2) + (8)(-4x)$$ Multiply those out: $$-7x -28 + 16 - 32x$$ Now simplify: $$-39x - 12$$ Summary: What Is the Distributive Property Definition? What is distributive property? The distributive property of multiplication states that $a(b+c) = ab + ac$. It’s often used for equations when the terms within the parentheses can’t be simplified because they contain one or more variables.Using the distributive property, you can simplify or solve equations that would otherwise be difficult to work with. When using the distributive property, remember to distribute negative signs if they’re in front of the parentheses, and keep in mind other important math rules, such as the quadratic formula, order of operations, and commutative properties. What's Next? Are you learning about logarithms and natural logs in math class? We have a guide on all the natural log rules you need to know. What is dynamic equilibrium and what does it have to do with rusty cars? Find out by reading ourcomplete guide to dynamic equilibrium. Rational numbers are another important math concept you should understand.Read our guide on rational numbers for everything you need to know about them!

The Roman Tetrarchy and the Rule of Four

The Roman Tetrarchy and the Rule of Four The word Tetrarchy means rule of four. It derives from the Greek words for four (tetra-) and rule (arch-). In practice, the word refers to the division of an organization or government into four parts, with a different person ruling each part. There have been several Tetrarchies over the centuries, but the phrase is usually used to refer to the division of the Roman Empire into a western and eastern empire, with subordinate divisions within the western and eastern empires. The Roman Tetrarchy Tetrarchy refers to the establishment by the Roman Emperor Diocletian of a 4-part division of the empire. Diocletian understood that the huge Roman Empire could be (and often was) taken over by any general who chose to assassinate the emperor. This, of course, caused significant political upheaval; it was virtually impossible to unite the empire. The reforms of Diocletian came after a period when many emperors had been assassinated. This earlier period is referred to as chaotic and the reforms were meant to remedy the political difficulties that the Roman Empire faced. Diocletians solution to the problem was to create multiple leaders, or Tetrarchs, located in multiple locations. Each would have significant power. Thus, the death of one of the Tetrarchs would not mean a change in governance. This new approach, in theory, would lower the risk of assassination and, at the same time, made it nearly impossible to overthrow the entire Empire at a single blow. When he split up the leadership of the Roman Empire in 286, Diocletian continued to rule in the East. He made Maximian his equal and co-emperor in the west. They were each called Augustus which signified that they were emperors. In 293, the two emperors decide to name additional leaders who could take over for them in the case of their deaths. Subordinate to the emperors were the two Caesars: Galerius, in the east, and Constantius in the west. An Augustus was always emperor; sometimes the Caesars were also referred to as emperors. This method of creating emperors and their successors bypassed the need for approval of emperors by the Senate and blocked the power of the military to elevate their popular generals to the purple. [Source: The City of Rome in late imperial ideology: The Tetrarchs, Maxentius, and Constantine, by Olivier Hekster, from Mediterraneo Antico 1999.] The Roman Tetrarchy functioned well during Diocletians life, and he and Maximian did indeed turn over leadership to the two subordinate Caesars, Galerius and Constantius. These two, in turn, named two new Caesars: Severus and Maximinus Daia.  The untimely death of Constantius, however, led to political warring. By 313, the Tetrarchy was no longer functional, and, in 324, Constantine became sole Emperor of Rome.   Other Tetrarchies While the Roman Tetrarchy is the most famous, other four-person ruling groups have existed through history. Among the best-known was The Herodian Tetrarchy, also called the Tetrarchy of Judea. This group, formed after the death of Herod the Great in 4 BCE, included Herods sons.