Solving Mathematical Expressions Using Properties
In the realm of mathematics, efficiency and elegance often go hand in hand. When faced with complex calculations, leveraging mathematical properties can significantly simplify the process and lead to accurate solutions. This article delves into the application of various properties to solve mathematical expressions, providing a step-by-step guide to mastering these techniques. We will explore how the distributive property, associative property, and commutative property can be strategically employed to streamline calculations and arrive at the desired results. Let's embark on this journey of mathematical exploration and discover the power of properties in simplifying complex expressions.
Understanding Mathematical Properties
Before we dive into solving specific problems, let's take a moment to review the fundamental mathematical properties that will be our guiding tools. These properties are the bedrock of efficient calculation and provide the justification for many of the techniques we will employ.
- Distributive Property: The distributive property is a cornerstone of simplifying expressions involving multiplication and addition (or subtraction). It states that for any numbers a, b, and c: a × (b + c) = a × b + a × c. This property allows us to distribute a common factor across a sum or difference, breaking down a complex multiplication into simpler terms. Conversely, we can also use the distributive property in reverse to factor out a common factor: a × b + a × c = a × (b + c).
- Associative Property: The associative property focuses on how numbers are grouped in addition or multiplication. It asserts that the grouping of numbers does not affect the outcome. For addition, it states that (a + b) + c = a + (b + c), and for multiplication, it states that (a × b) × c = a × (b × c). This property provides the flexibility to rearrange terms in an expression to make calculations more convenient.
- Commutative Property: The commutative property addresses the order of numbers in addition or multiplication. It states that the order in which numbers are added or multiplied does not alter the result. For addition, a + b = b + a, and for multiplication, a × b = b × a. This property allows us to rearrange terms to group similar numbers together or to facilitate easier calculations.
With these properties firmly in mind, we are now equipped to tackle a series of mathematical expressions and witness their power in action.
Applying Properties to Simplify Expressions
Now, let's put our knowledge of mathematical properties into practice by solving the given expressions. We will systematically apply the distributive, associative, and commutative properties to simplify each expression and arrive at the solution.
(i) 647 × 13 + 647 × 7
In this expression, we observe a common factor of 647 in both terms. This immediately suggests the application of the distributive property in reverse, also known as factoring. By factoring out 647, we can transform the expression into a more manageable form.
Step 1: Identify the common factor
The common factor in this expression is 647.
Step 2: Apply the distributive property (factoring)
We can rewrite the expression as follows:
647 × 13 + 647 × 7 = 647 × (13 + 7)
Step 3: Simplify the expression within the parentheses
13 + 7 = 20
Step 4: Perform the final multiplication
647 × 20 = 12940
Therefore, the value of the expression 647 × 13 + 647 × 7 is 12940. This example beautifully illustrates how the distributive property can streamline calculations by consolidating terms and reducing the number of operations.
(ii) 8759 × 94 + 8759 × 6
Similar to the previous example, this expression also presents a common factor, making the distributive property our primary tool for simplification. By recognizing and factoring out the common factor, we can significantly reduce the complexity of the calculation.
Step 1: Identify the common factor
The common factor in this expression is 8759.
Step 2: Apply the distributive property (factoring)
We can rewrite the expression as:
8759 × 94 + 8759 × 6 = 8759 × (94 + 6)
Step 3: Simplify the expression within the parentheses
94 + 6 = 100
Step 4: Perform the final multiplication
8759 × 100 = 875900
Thus, the value of the expression 8759 × 94 + 8759 × 6 is 875900. The distributive property, in this case, transforms a potentially lengthy calculation into a simple multiplication by 100, highlighting its efficiency.
(iii) 7459 × 999 + 7459
This expression might appear slightly different at first glance, but a closer look reveals a hidden common factor. We can rewrite the term 7459 as 7459 × 1 to make the common factor more apparent. This subtle adjustment allows us to apply the distributive property effectively.
Step 1: Rewrite the expression to reveal the common factor
7459 × 999 + 7459 = 7459 × 999 + 7459 × 1
Step 2: Identify the common factor
The common factor is now clearly 7459.
Step 3: Apply the distributive property (factoring)
We can rewrite the expression as:
7459 × 999 + 7459 × 1 = 7459 × (999 + 1)
Step 4: Simplify the expression within the parentheses
999 + 1 = 1000
Step 5: Perform the final multiplication
7459 × 1000 = 7459000
Therefore, the value of the expression 7459 × 999 + 7459 is 7459000. By recognizing the implicit multiplication by 1 and applying the distributive property, we simplified the calculation significantly.
(iv) 9870 × 561 - 9870 × 461
This expression involves subtraction, but the principle of applying the distributive property remains the same. We identify the common factor and factor it out, transforming the subtraction into a simpler operation within parentheses.
Step 1: Identify the common factor
The common factor in this expression is 9870.
Step 2: Apply the distributive property (factoring)
We can rewrite the expression as:
9870 × 561 - 9870 × 461 = 9870 × (561 - 461)
Step 3: Simplify the expression within the parentheses
561 - 461 = 100
Step 4: Perform the final multiplication
9870 × 100 = 987000
Thus, the value of the expression 9870 × 561 - 9870 × 461 is 987000. The distributive property efficiently reduces the calculation to a single multiplication by 100.
(v) 569 × 17 + 569 × 13 + 569 × 70
This expression extends the concept of the distributive property to three terms, but the underlying principle remains consistent. We identify the common factor and factor it out, combining the remaining terms within parentheses.
Step 1: Identify the common factor
The common factor in this expression is 569.
Step 2: Apply the distributive property (factoring)
We can rewrite the expression as:
569 × 17 + 569 × 13 + 569 × 70 = 569 × (17 + 13 + 70)
Step 3: Simplify the expression within the parentheses
17 + 13 + 70 = 100
Step 4: Perform the final multiplication
569 × 100 = 56900
Therefore, the value of the expression 569 × 17 + 569 × 13 + 569 × 70 is 56900. The distributive property efficiently consolidates multiple multiplications into a single, straightforward calculation.
(vi) 16825 × 16825 - 16825 × 6825
In this expression, we again encounter a common factor, but the numbers are larger, making the application of the distributive property even more crucial for simplification. Factoring out the common factor transforms the expression into a more manageable form.
Step 1: Identify the common factor
The common factor in this expression is 16825.
Step 2: Apply the distributive property (factoring)
We can rewrite the expression as:
16825 × 16825 - 16825 × 6825 = 16825 × (16825 - 6825)
Step 3: Simplify the expression within the parentheses
16825 - 6825 = 10000
Step 4: Perform the final multiplication
16825 × 10000 = 168250000
Thus, the value of the expression 16825 × 16825 - 16825 × 6825 is 168250000. The distributive property, in this case, transforms a complex subtraction and multiplication into a simpler multiplication by 10000, demonstrating its power in handling larger numbers.
Determine Each of Discussion Category
The discussion category for the problems presented in this article falls under the domain of mathematics, specifically within the subcategories of arithmetic and algebra. The problems primarily involve numerical calculations and the application of fundamental mathematical properties, such as the distributive property, to simplify expressions. These concepts are foundational to mathematical understanding and are typically introduced in early stages of mathematical education. The problems encourage the development of algebraic thinking by emphasizing the manipulation of expressions and the identification of patterns and common factors.
In this article, we have explored the power of mathematical properties in simplifying complex expressions. By strategically applying the distributive property, we have transformed seemingly daunting calculations into manageable steps. The ability to recognize common factors and factor them out is a crucial skill in mathematics, enabling efficient problem-solving and a deeper understanding of mathematical relationships. As you continue your mathematical journey, remember to leverage these properties to streamline your calculations and unlock the beauty of mathematical simplicity. Embracing these techniques will not only enhance your problem-solving abilities but also foster a more profound appreciation for the elegance and efficiency inherent in the world of mathematics.