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What are properties of addition and multiplication?
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Properties are statements that are true for all numbers.
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First up, commutative properties.
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Weβll start with the commutative property of addition.
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The commutative property of addition states that the order in which two numbers are added does not change their sum.
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A variable representation of that is π plus π is the same thing as saying π plus π.
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Or four plus three is the same thing as three plus four.
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The order doesnβt matter here.
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The commutative property can also be applied to multiplication.
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Look at the changes that happened on the screen.
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Commutative property of multiplication states that the order in which two numbers are multiplied does not change their product.
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In this case, π times π equals π times π.
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An example of that is five times four equals four times five.
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One way to remember the commutative property is to think about the word commute.
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The word commute and the word commutative relates to exchange, substitution, and interchange.
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Here with the commutative property, weβre specifically talking about when we change the order in which we add or multiply.
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And in this case, it does not change the value depending on the order.
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Next up, the associative property.
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And again weβll start with addition.
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The associative property of addition states the way three numbers are grouped when they are added does not change their sum.
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Hereβs what that looks like with variables.
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π plus π plus π equals π plus π plus π.
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What that means is if we add π and π together first and then add π, that sum is the same as if we added π and π together first and then added π.
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In this case, one plus two plus three equals one plus two plus three.
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One plus five equals three plus three.
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This property can also be applied to multiplication.
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Notice the changes here.
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The associative property of multiplication says the way three numbers are grouped when they are multiplied does not change their product.
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If we multiply π and π first and then take that value and multiply it by π, we will have the same product as if weβve multiplied π and π and then that value by π.
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We can remember the associative property with the word associate.
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Association deals with groupings.
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The way that three numbers are grouped when theyβre added or multiplied does not change their sum or their product.
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And third on our list, identity properties.
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Starting with the identity property of addition, this property states that the sum of an addend and zero is the addend.
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π plus zero equals π.
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Seven plus zero is seven.
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As we switch over to the identity property of multiplication, letβs look carefully at all the changes.
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The identity property of multiplication says the product of a factor and one is that factor.
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Our example, π times one equals π.
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Notice that the identity property of multiplication is multiplying the factor by one.
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Letβs look at these properties side by side.
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When we add zero to any value, weβre going to get the same value back.
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And when we multiply by one, the same thing happens.
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You can remember the identity property by thinking about looking in the mirror.
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π looks the same after you add zero.
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π also looks the same after you multiply it by one.
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Our last property is a little bit different.
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The distributive property shows us how we combine addition and multiplication.
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The distributive property says this.
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To multiply a sum by a number, multiply each addend of the sum by the number outside the parentheses.
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Thatβs a lot of words.
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Letβs see what that looks like.
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Letβs start by looking at an example that uses numbers.
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The property tells us to multiply a sum by a number.
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Hereβs our sum and hereβs the number.
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We need to multiply each addend of the sum.
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Four and six are the addends of the sum.
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And we multiply that by the number outside the parentheses, in this case, the three.
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If you solve both sides of the equation, you end up with thirty equals thirty.
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The distributive property helps us take this three and distribute it across the four and the six, the two addends of the sum.
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An algebra representation of the distributive property or a representation with variables would look like this.
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π times π plus π equals π times π plus π times π.
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What weβre doing here is weβre taking the π and weβre distributing it across the addend π and the addend π.
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This is also true in reverse.
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This is also true in reverse.
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In this example, we wanna take the π out, put it outside the parentheses and add the π and π first.
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Hereβs that example with numbers.
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Five times two plus five times three is the same thing as saying five times five or five times two plus three.
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Hereβs a chart to help you summarize all the different properties.
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These properties will be the foundation of solving all kinds of equations.