Distributive Property – Definition, FAQs, Examples

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    The distributive property is a fundamental mathematical idea that helps make complicated algebraic expressions easier to understand. It lets scholars manipulate arithmetic operations, making complicated equations easier to solve. In greater depth, we will investigate the distributive property’s uses and applications, as well as distributive property examples in this article. But first, what is distributive property?

    What Is Distributive Property?

    The distributive property definition often depends on the numerous contexts of the math concept application. For example, it allows mathematicians to distribute multiplication over addition or subtraction. This law is crucial and based on equality. It simplifies and manipulates expressions, making them easier to solve.

    You can distribute a multiplication operation over subtraction in the same way you distribute a multiplication operation over addition when the distributive property is applied to multiplication and subtraction. This regulation means that the duplication of a number by the distinction of at least two numbers is equivalent to the distinction of the results of that number and each term in the distinction.

    What does distributive property mean in algebraic expressions where it is essential to simplify and manipulate expressions to solve complex equations? This math concept allows us to simplify expressions by factoring, expanding, and combining similar terms. Moreover, it can assist us in addressing conditions with factors by streamlining and controlling the expressions containing those factors.

    Distributive Property Formula

    The distributive property formula can be written as follows: Where a, b, and c are real numbers, a(b+c) equals ab + ac, and a(b-c) equals ab – ac.

    To put it another way, if we have a number “a” multiplied by the sum or difference of two other numbers, “b” and “c,” we can divide “a’s” multiplication over each of these numbers individually and then add or subtract the products.

    Distributive Property of Multiplication over Addition

    The distributive property of multiplication over addition is different from the distributive property of addition in that the sum of the products of a number and any two or more terms is the same as the sum of the products of the number and each term in the sum. For instance, if we have the expression 2(x + y), we can divide two by the sum of x and y to get the expression 2x + 2y. By employing the distributive property, we have reduced the original expression into two simpler terms, which could have been challenging.

    Distributive Property of Multiplication over Subtraction

    The distributive property of multiplication over subtraction also states that the difference between a number’s product and between two or more terms is the same as the difference between the number’s product and each term in the difference. So, for example, if we have the expression 3(x – y), we can disperse the expansion of 3 over the difference of x and y, bringing about the expression 3x – 3y. Once more, we have worked on the first expression into two less complex terms, making it much simpler to settle and work with.

    This type of distributive property is generally utilized in polynomial math as it permits us to improve on expressions that include different terms. By dispersing the increase over each term, we can separate complex expressions into less difficult ones, which diminishes the number of terms and simplifies the expression. Furthermore, we can ensure that our calculations are error-free and that the result is correct by verifying the distributive property after simplification.

    Verification of Distributive Property

    You can’t claim to know how to use distributive property without understanding how to verify it. Verifying the distributive property is a stage that guarantees that the property has been applied accurately and that the two sides of a situation are the same. This procedure requires performing the calculation on both sides to verify that the two sides of the equation are equal.

    To test the distributive property, first multiply each term within the parentheses by the number outside the parentheses, which is also known as the distributive factor. You should apply the multiplication to each term separately if multiple terms are contained within the parentheses. Depending on whether addition or subtraction preceded the parentheses, you should then add or subtract the resulting products. This step disseminates the expansion or division activity over each term inside the parentheses as per the distributive property.

     The original expressions on both sides of the equation are then compared to the calculation result. The distributive property was correctly used if both sides of the equation are the same and the results are the same.

    For instance, 4(x + 3) = 4x + 12. 4x + 12 can be obtained by multiplying 4 by 3 and 4 by x to confirm the distributive property. You can see that this result and the original expression 4(x + 3) are the same. Therefore, the distributive property was correctly utilized in this instance.

    It is essential to verify the distributive property to avoid making mistakes when solving equations and guarantee the accuracy of mathematical calculations. It is a crucial step in algebraic manipulations to simplify expressions and solve equations, which provides a means to verify that the distributive property has been applied correctly. We can ensure the accuracy of our mathematical work and establish a solid foundation for further mathematical reasoning and problem-solving by examining the distributive property.

    Distributive Property of Division

    We can distribute the division operation over each term within the parentheses and then divide the result by the same number when dividing a number by the sum or difference of two or more terms. We can add or subtract the results of dividing a number by each term in the parentheses separately. For instance, let’s think about articulation 24/(6 + 2). The expression 24/6 + 24/2 can be obtained by dividing 24 over 6 and 2. It works for the expression 4 + 12, which offers us the answer 16.

     Similarly, we can divide the result by the same number after distributing the division operation over each term in the difference when dividing a number by the difference of two or more terms. For example, consider the expression 16/(8 – 2). The expression 16/8 – 16/2 can be obtained by distributing the division of 16 over the difference between 8 and 2. The expression is reduced to 2-8 because of this, and our final result is -6.

    Division’s distributive property can also be used to simplify expressions with more terms, which is more complicated. Consider, for instance, the example 30/(5 + 3 + 2). First, the expression 30/5 + 30/3 + 30/2 can be obtained by dividing 30 by the sum of 5, 3, and 2. Then, the expression is reduced to 6 + 10 + 15, resulting in the answer 31.

    When employing the distributive property of division, it is essential to remember that the order of operations remains in effect. It implies we should perform any tasks inside the enclosures before appropriating the division over each term. The next section will show examples of how to apply distributive property.

    Solved Distributive Property Examples

    Example 1

    Make the expression simpler: 3(2x + 5). 

    Answer:

    3(2x + 5) = 3 x 2x + 3 x 5 = 6x + 15.

    This way, the expression is 6x + 15.

    Example 2

    Make the expression simpler: 4(3a – 2b). 

    Answer:

    4(3a – 2b) = 4 x 3a – 4 x 2b = 12a – 8b

    Subsequently, the expression becomes 12a – 8b.

    Example 3

    Make the expression simpler: 2(4x – 3) – 3(x + 2). 

    Answer

    Therefore, the simplified expression is 5x – 12: 2(4x – 3) – 3(x + 2) = 8x – 6 – 3x – 6.

    Frequently Asked Questions

    Does distributive property apply to division too?

    Indeed, the distributive property applies to division also. We can distribute the division operation over the terms within the parenthesis when dividing a number by the sum or difference of two or more numbers and then divide the result by the same number.

    What is the rule for the distributive property?

    The standard for the distributive property is to increase or gap the number outside the bracket with each term inside the bracket and afterward add or take away the items. It aids in improving complex expressions and addressing conditions.

    How can distributive property help in solving complex questions?

    The distributive property makes it easier to solve a complicated expression by simplifying it. We can reduce the number of terms and simplify the expression by utilizing the distributive property. It, in turn, makes equations easier to solve and calculations easier to perform.

    What Is the Distributive Property for Rational Numbers?

    The distributive property applies to a wide range of numbers, including objective numbers. For example, we can use the distributive property to get the expression 2/3(3x + 6) by multiplying the fraction outside the brackets by each term inside the brackets. 2x + 4 will be the result.

    Where Is the Distributive Property Used?

    Algebra, calculus, and geometry are just a few mathematical disciplines that use the distributive property. It is also utilized in engineering, physics, and chemistry to solve problems involving complex expressions and equations.

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