Simplifying Expressions Using The Distributive Property 1/3(33c+12)
The distributive property is a fundamental concept in mathematics that allows us to simplify expressions involving multiplication and addition or subtraction. It's a powerful tool that helps us break down complex expressions into simpler, more manageable forms. In this article, we will delve into the distributive property, explore its applications, and demonstrate how to use it effectively to simplify expressions. We will specifically focus on simplifying the expression $ rac{1}{3}(33 c+12)$ using the distributive property, providing a step-by-step explanation to ensure a clear understanding of the process. This comprehensive guide aims to equip you with the knowledge and skills necessary to confidently apply the distributive property in various mathematical contexts.
At its core, the distributive property states that multiplying a sum or difference by a number is the same as multiplying each term inside the parentheses by that number and then adding or subtracting the results. Mathematically, this can be expressed as follows:
- a( b + c) = a b + a c
- a( b - c) = a b - a c
Where a, b, and c represent any real numbers. The distributive property is a cornerstone of algebra and is crucial for simplifying expressions, solving equations, and performing various mathematical operations. Mastering this property is essential for building a strong foundation in mathematics.
To illustrate the distributive property, let's consider a simple example: 2( x + 3). According to the distributive property, we can multiply 2 by both x and 3 separately and then add the results. This gives us 2 * x + 2 * 3, which simplifies to 2x + 6. This example demonstrates how the distributive property allows us to eliminate parentheses and simplify expressions.
The distributive property is not limited to simple expressions; it can be applied to more complex expressions involving multiple terms and variables. For instance, consider the expression 4(2x - 5y + 1). Applying the distributive property, we multiply 4 by each term inside the parentheses: 4 * 2x - 4 * 5y + 4 * 1, which simplifies to 8x - 20y + 4. This example showcases the versatility of the distributive property in handling various types of expressions.
The distributive property is closely related to the concept of factoring. Factoring is the reverse process of distribution, where we identify a common factor in an expression and factor it out. For example, consider the expression 6x + 9. We can see that both terms have a common factor of 3. Factoring out 3, we get 3(2x + 3). Understanding the relationship between distribution and factoring is crucial for simplifying expressions and solving equations.
In summary, the distributive property is a fundamental principle in mathematics that allows us to simplify expressions involving multiplication and addition or subtraction. It states that multiplying a sum or difference by a number is the same as multiplying each term inside the parentheses by that number and then adding or subtracting the results. Mastering this property is essential for success in algebra and beyond.
Now, let's apply the distributive property to simplify the expression $ rac{1}{3}(33 c+12)$. We will break down the process into clear, concise steps to ensure a thorough understanding.
Step 1: Identify the terms inside the parentheses and the number outside the parentheses.
In the given expression, $ rac{1}{3}(33 c+12)$, the terms inside the parentheses are 33c and 12, and the number outside the parentheses is $ rac{1}{3}$.
Step 2: Apply the distributive property by multiplying the number outside the parentheses by each term inside the parentheses.
According to the distributive property, we need to multiply $ rac{1}{3}$ by both 33c and 12. This gives us:
rac{1}{3} * 33c + rac{1}{3} * 12
Step 3: Perform the multiplication.
Now, let's perform the multiplication:
rac{1}{3} * 33c = 11c
rac{1}{3} * 12 = 4
Step 4: Combine the results.
Finally, we combine the results from the previous step:
Therefore, the simplified expression is 11c + 4. This step-by-step process demonstrates how the distributive property can be effectively used to simplify expressions involving fractions and variables.
To further solidify your understanding, let's consider another example: $ rac1}{2}(10x - 6)$. Applying the distributive property, we multiply $ rac{1}{2}$ by both 10x and -6. This gives us{2} * 10x - rac{1}{2} * 6$, which simplifies to 5x - 3. This example reinforces the application of the distributive property with different coefficients and constants.
In summary, simplifying the expression $ rac{1}{3}(33 c+12)$ using the distributive property involves multiplying $ rac{1}{3}$ by both 33c and 12, which results in 11c + 4. This step-by-step approach provides a clear and concise method for applying the distributive property to simplify expressions.
To further solidify your understanding of the distributive property, let's explore several illustrative examples. These examples will demonstrate how to apply the distributive property in various scenarios, including expressions with different coefficients, variables, and operations.
Example 1: Simplify the expression 5( x + 4).
Applying the distributive property, we multiply 5 by both x and 4:
5 * x + 5 * 4 = 5x + 20
Therefore, the simplified expression is 5x + 20. This example demonstrates a basic application of the distributive property with whole numbers.
Example 2: Simplify the expression -2(3y - 7).
Applying the distributive property, we multiply -2 by both 3y and -7:
-2 * 3y - (-2) * 7 = -6y + 14
Therefore, the simplified expression is -6y + 14. This example highlights the importance of paying attention to signs when applying the distributive property.
Example 3: Simplify the expression $ rac{2}{3}(9z + 15)$.
Applying the distributive property, we multiply $ rac{2}{3}$ by both 9z and 15:
rac{2}{3} * 9z + rac{2}{3} * 15 = 6z + 10
Therefore, the simplified expression is 6z + 10. This example demonstrates the application of the distributive property with fractions.
Example 4: Simplify the expression a( b + c - d).
Applying the distributive property, we multiply a by b, c, and -d:
a * b + a * c - a * d = ab + ac - ad
Therefore, the simplified expression is ab + ac - ad. This example showcases the distributive property with multiple variables.
Example 5: Simplify the expression 2x( x - 3).
Applying the distributive property, we multiply 2x by both x and -3:
2x * x - 2x * 3 = 2x2 - 6x
Therefore, the simplified expression is 2x2 - 6x. This example demonstrates the distributive property with variables and exponents.
These examples illustrate the versatility of the distributive property in simplifying various types of expressions. By consistently applying the distributive property, you can effectively simplify complex expressions and solve mathematical problems.
While the distributive property is a powerful tool, it's essential to be aware of common mistakes that can occur when applying it. By understanding these pitfalls, you can avoid errors and ensure accurate simplification of expressions.
1. Forgetting to Distribute to All Terms:
One of the most common mistakes is failing to distribute the number outside the parentheses to all terms inside the parentheses. For example, in the expression 2( x + 3), some might incorrectly multiply 2 only by x, resulting in 2x + 3. The correct application of the distributive property requires multiplying 2 by both x and 3, resulting in 2x + 6.
2. Incorrectly Handling Signs:
Another frequent error involves incorrectly handling signs, especially when dealing with negative numbers. For example, in the expression -3(2y - 5), some might incorrectly distribute the -3 as -3 * 2y - 5, which simplifies to -6y - 5. The correct application of the distributive property involves multiplying -3 by both 2y and -5, resulting in -6y + 15.
3. Misapplying the Distributive Property with Exponents:
The distributive property applies to multiplication over addition or subtraction, not to exponents. For example, ( x + y)2 is not equal to x2 + y2. The correct way to expand ( x + y)2 is to multiply ( x + y) by itself: (x + y)(x + y), which then requires the application of the distributive property (often referred to as the FOIL method).
4. Confusing Distribution with Order of Operations:
The distributive property should be applied before performing addition or subtraction outside the parentheses. For example, in the expression 4 + 2( x + 1), some might incorrectly add 4 and 2 first, resulting in 6( x + 1). The correct approach is to first distribute the 2: 4 + 2x + 2, and then combine like terms: 2x + 6.
5. Applying the Distributive Property in the Wrong Context:
The distributive property is specifically for multiplication over addition or subtraction. It does not apply to other operations, such as division. For example, $ racx + y}{z}$ is not equal to $ rac{x}{z} + y$. To correctly divide a sum or difference by a number, you must divide each term by that number{z} = rac{x}{z} + rac{y}{z}$.
By being mindful of these common mistakes, you can enhance your accuracy and proficiency in applying the distributive property.
The distributive property is not just a theoretical concept confined to textbooks; it has numerous real-world applications in various fields. Understanding these applications can help you appreciate the practical significance of this mathematical principle.
1. Calculating Costs and Discounts:
The distributive property is frequently used in everyday financial calculations. For instance, when calculating the total cost of multiple items with a discount, the distributive property can simplify the process. Suppose you want to buy 3 shirts that cost $20 each, and there's a 10% discount on the total purchase. You can calculate the discounted price using the distributive property:
Total cost before discount: 3 * $20 = $60
Discount amount: 10% of $60 = 0.10 * $60 = $6
Discounted price: $60 - $6 = $54
Alternatively, you can use the distributive property:
Discounted price: (1 - 0.10) * (3 * $20) = 0.90 * $60 = $54
This demonstrates how the distributive property can streamline calculations involving discounts and costs.
2. Measuring Areas and Volumes:
The distributive property is also applied in geometry when calculating areas and volumes. For example, consider a rectangular garden with a length of ( x + 5) meters and a width of 4 meters. The area of the garden can be calculated using the distributive property:
Area = width * length = 4 * (x + 5) = 4x + 20 square meters
Similarly, the distributive property can be used to calculate the volume of a composite shape, such as a rectangular prism with a smaller rectangular prism removed.
3. Simplifying Algebraic Expressions in Science and Engineering:
In science and engineering, algebraic expressions often arise in various formulas and equations. The distributive property is crucial for simplifying these expressions and solving for unknown variables. For example, in physics, the formula for the force exerted by a spring is F = -k(x - x0), where F is the force, k is the spring constant, x is the displacement, and x0 is the equilibrium position. To analyze this equation, the distributive property can be applied to expand the expression: F = -kx + kx0.
4. Computer Programming:
In computer programming, the distributive property is used in various algorithms and data structures. For example, when manipulating arrays or matrices, the distributive property can be applied to optimize calculations and improve efficiency. Additionally, the distributive property is used in symbolic computation and computer algebra systems to simplify complex expressions.
These examples illustrate the diverse real-world applications of the distributive property. From everyday financial calculations to complex scientific and engineering problems, the distributive property is a fundamental tool for simplifying expressions and solving problems.
In conclusion, the distributive property is a fundamental concept in mathematics with wide-ranging applications. It allows us to simplify expressions involving multiplication and addition or subtraction, making complex problems more manageable. By understanding the distributive property and practicing its application, you can enhance your mathematical skills and problem-solving abilities.
In this article, we have explored the distributive property in detail, providing a step-by-step guide to simplifying expressions. We have also examined common mistakes to avoid and discussed real-world applications of the distributive property. By mastering this essential mathematical principle, you will be well-equipped to tackle a variety of mathematical challenges.
Remember, the key to success in mathematics is consistent practice and a thorough understanding of fundamental concepts. The distributive property is a cornerstone of algebra, and its mastery will pave the way for more advanced mathematical studies. So, continue to practice, explore, and apply the distributive property in various contexts to solidify your understanding and unlock its full potential.
