Definition of distributive property and Application of distributive property in mathematics

Definition of distributive property

The distributive property is a mathematical property that allows you to combine numbers and operations in a certain way. It states that when multiplying a number by the sum or difference of two or more other numbers, you can distribute the multiplication to each term inside the parentheses.

In algebra, the distributive property can be expressed as:

a × (b + c) = (a × b) + (a × c)

This means that if you have a number ‘a’ multiplied by the sum of ‘b’ and ‘c’, you can distribute the multiplication to each term, resulting in the product of ‘a’ and ‘b’ added to the product of ‘a’ and ‘c’.

Similarly, the distributive property also applies to subtraction:

a × (b – c) = (a × b) – (a × c)

This means that if you have a number ‘a’ multiplied by the difference of ‘b’ and ‘c’, you can distribute the multiplication to each term, resulting in the product of ‘a’ and ‘b’ subtracted by the product of ‘a’ and ‘c’.

The distributive property is a fundamental concept in algebra and is used to simplify and solve equations, as well as to manipulate algebraic expressions.

Application of distributive property in mathematics

The distributive property is a fundamental concept in mathematics that is applied in various areas. It allows us to simplify expressions and perform operations efficiently. Here are some common applications of the distributive property:

1. Simplifying Expressions: The distributive property is commonly used to simplify algebraic expressions. It states that for any three numbers a, b, and c, the expression a(b + c) is equivalent to ab + ac. This property helps in removing parentheses and combining like terms in order to simplify an expression.

Example: Simplify the expression 3(2x + 4y) + 5z.

Using the distributive property, we can expand the expression as:

= 3 * 2x + 3 * 4y + 5z

= 6x + 12y + 5z

2. Multiplying Numbers: The distributive property is also used when multiplying numbers that have more than one digit. It allows us to break down the multiplication into smaller, simpler operations.

Example: Multiply 23 by 8.

Using the distributive property, we can rewrite 23 as 20 + 3, and then distribute the 8 to each term:

= 8 * 20 + 8 * 3

= 160 + 24

= 184

3. Factoring: The distributive property is used when factoring algebraic expressions. It helps in decomposing expressions into their factors, which is useful in solving equations, finding common factors, and simplifying problems.

Example: Factor the expression 6x + 18.

Using the distributive property, we can rewrite 6x + 18 as 6(x + 3), where 6 is a common factor:

= 6(x + 3)

4. Finding the Area of Shapes: The distributive property is applied to find the area of complex shapes by breaking them down into simpler components. It helps in calculating the total area by multiplying the dimensions of each part.

Example: Find the area of the figure below:

+—+—+

| |

+—+—+

| |

+—+—+

This figure consists of two rectangles. To find the total area, we can apply the distributive property:

= (2 * 4) + (2 * 2)

= 8 + 4

= 12 square units

Overall, the distributive property is a versatile tool that is used extensively in algebra, arithmetic, and geometry to simplify expressions, solve problems, and manipulate equations effectively.

Examples of distributive property

Here are some examples of the distributive property in action:

1. Example: 3 * (4 + 2)

Using the distributive property, we can distribute the 3 to both the 4 and the 2:

3 * 4 + 3 * 2

This simplifies to:

12 + 6 = 18

2. Example: 2 * (5 – 3)

Using the distributive property, we distribute the 2 to both the 5 and the -3:

2 * 5 – 2 * 3

This simplifies to:

10 – 6 = 4

3. Example: 7 * (a + b)

Using the distributive property, we distribute the 7 to both the a and the b:

7 * a + 7 * b

This cannot be simplified any further, so the answer is:

7a + 7b

4. Example: (x – 2) * (3 + y)

Using the distributive property, we distribute the (x – 2) to both the 3 and the y:

x * 3 + x * y – 2 * 3 – 2 * y

This simplifies to:

3x + xy – 6 – 2y

Importance of understanding distributive property

The distributive property is a fundamental concept in mathematics that is used to simplify and solve mathematical expressions and equations. It states that multiplying a number by a sum or difference is the same as multiplying the number separately by each term in the sum or difference, and then adding or subtracting the results.

Understanding the distributive property is important for several reasons:

1. Simplifying equations: By applying the distributive property, you can simplify complex equations or expressions into simpler forms. This makes it easier to manipulate and solve the equation.

2. Solving equations: The distributive property is often used to solve linear equations. It allows you to expand parentheses and combine like terms, which helps in isolating the variable and finding its value.

3. Factoring expressions: The distributive property is also used in reverse to factorize expressions. By factoring out a common term or factor, you can simplify an expression and write it in a more compact or organized form.

4. Algebraic Manipulation: The distributive property is a key tool in algebraic manipulations. It enables you to perform various operations, such as adding and subtracting polynomials, combining like terms, and solving quadratic equations.

5. Understanding mathematical concepts: Understanding the distributive property helps in understanding other important mathematical concepts, such as the associative and commutative properties. It provides a foundation for more advanced topics in algebra and calculus.

Overall, the distributive property is a fundamental concept in mathematics that is used in a wide range of applications. It allows for simplifying and solving equations, factoring expressions, and manipulating algebraic expressions. Therefore, comprehending and applying the distributive property is crucial for success in mathematics.

Further resources for learning about distributive property

Here are some further resources for learning about the distributive property:

1. Video tutorials: YouTube has numerous videos that explain the concept of the distributive property. Some popular channels include Khan Academy, Math Antics, and Math is Fun.

2. Online lessons: Websites like MathisFun.com, Mathplanet.com, and IXL.com offer interactive lessons on the distributive property. They provide step-by-step explanations, practice problems, and quizzes to test your understanding.

3. Worksheets and practice problems: Websites such as Math-Drills.com and SuperTeacherWorksheets.com have printable worksheets that focus on the distributive property. These can be a great way to practice applying the concept to different problems.

4. Math textbooks: Many math textbooks have dedicated sections on the distributive property. Check your local library or bookstore for textbooks that cover this topic. Some popular options include “Prentice Hall Mathematics: Course 2” and “Holt McDougal Larson Algebra 1.”

5. Online games and activities: Websites like Funbrain.com and MathPlayground.com offer interactive games and activities that help reinforce the distributive property. These games can make learning more engaging and enjoyable.

6. Online forums and communities: Joining online forums or communities dedicated to math education can be a valuable resource. Websites like MathHelpForum.com and Reddit’s r/learnmath subreddit allow you to ask questions and discuss the distributive property with other learners and experts.

Remember, practice is key when it comes to understanding the distributive property. Try to solve a variety of problems and explore different scenarios to solidify your understanding of this important mathematical concept.

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