Equivalent Expressions For 4(4a + 5) - Math Problem Solution

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In the realm of mathematics, particularly algebra, the concept of equivalent expressions is fundamental. Equivalent expressions, simply put, are expressions that, despite appearing different, yield the same value for all possible values of the variable involved. Identifying and manipulating equivalent expressions is a crucial skill for simplifying equations, solving problems, and gaining a deeper understanding of mathematical relationships. This article delves into the process of determining expressions equivalent to a given expression, using the example of 4(4a + 5). We will explore various techniques, including the distributive property and combining like terms, to unveil the expressions that hold the same mathematical value. Understanding equivalent expressions not only strengthens your algebraic foundation but also enhances your problem-solving capabilities in various mathematical contexts.

Understanding the Core Expression: 4(4a + 5)

To embark on our journey of identifying equivalent expressions, we must first dissect the given expression: 4(4a + 5). This expression showcases a fundamental algebraic structure: a constant multiplied by a binomial. The binomial, (4a + 5), represents the sum of two terms: 4a, a variable term where 'a' is multiplied by the coefficient 4, and 5, a constant term. The constant 4 outside the parentheses acts as a multiplier for the entire binomial. To effectively work with this expression and find its equivalents, we need to apply the distributive property, a cornerstone of algebraic manipulation. The distributive property dictates that multiplying a sum (or difference) by a number is the same as multiplying each term of the sum (or difference) individually by that number and then adding (or subtracting) the products. In simpler terms, we distribute the outer multiplier to each term inside the parentheses. This initial step of applying the distributive property is crucial for expanding the expression and revealing its underlying structure, paving the way for identifying its equivalent forms. Without a solid grasp of the distributive property, navigating the world of algebraic expressions becomes significantly more challenging. It's a tool that empowers us to simplify complex expressions and transform them into more manageable forms, ultimately aiding in solving equations and unraveling mathematical puzzles.

Applying the Distributive Property

The distributive property is the key to unlocking equivalent expressions for 4(4a + 5). This property, a cornerstone of algebra, allows us to multiply the term outside the parentheses (in this case, 4) by each term inside the parentheses individually. Let's break down the application of the distributive property step-by-step:

  1. Multiply 4 by 4a: 4 * 4a = 16a. This step involves multiplying the constant 4 by the variable term 4a. Remember that when multiplying a constant by a variable term, we multiply the coefficients (the numerical parts) together. In this case, 4 multiplied by 4 equals 16, and the variable 'a' remains unchanged.
  2. Multiply 4 by 5: 4 * 5 = 20. Here, we multiply the constant 4 by the constant term 5. This is a straightforward multiplication that results in 20.
  3. Combine the results: Now, we combine the results from the previous two steps: 16a + 20. This step is crucial as it brings together the individual products obtained from the distribution, forming a new expression.

Therefore, applying the distributive property to 4(4a + 5) yields the expression 16a + 20. This resulting expression is a direct equivalent of the original, meaning it holds the same value for any value of 'a'. The distributive property essentially unpacks the expression, making it easier to compare with other potential equivalent forms. Understanding and mastering this property is fundamental for simplifying algebraic expressions and solving equations efficiently. It acts as a bridge, connecting seemingly different forms of expressions and allowing us to manipulate them effectively.

Evaluating the Answer Choices

Now that we've simplified the original expression 4(4a + 5) to its equivalent form 16a + 20, we can evaluate the provided answer choices to determine which ones match. This process involves comparing each choice with our simplified expression and applying algebraic principles to verify their equivalence. Let's examine each answer choice systematically:

A. 16a + 5

This expression, 16a + 5, appears similar to our simplified form 16a + 20, but a crucial difference lies in the constant term. While both expressions have the same variable term (16a), the constant term differs (5 versus 20). This difference signifies that the two expressions are not equivalent. For expressions to be equivalent, they must have the same terms with the same coefficients. In this case, the differing constant terms indicate that the expressions will yield different values for the same value of 'a'. Therefore, 16a + 5 is not equivalent to 4(4a + 5).

B. 16a + 20

Upon immediate comparison, we see that this expression, 16a + 20, is identical to the simplified form we derived earlier using the distributive property. Both expressions have the same variable term (16a) and the same constant term (20). This direct match confirms that 16a + 20 is indeed an equivalent expression to 4(4a + 5). The equivalence is clear and requires no further manipulation or simplification. This choice demonstrates the direct application of the distributive property, showcasing how a seemingly different expression can be transformed into its equivalent form through algebraic manipulation.

C. 12a + 20 + 4a

This expression, 12a + 20 + 4a, presents an opportunity to utilize another essential algebraic technique: combining like terms. Like terms are terms that have the same variable raised to the same power. In this expression, 12a and 4a are like terms because they both contain the variable 'a' raised to the power of 1. To simplify the expression, we can combine these like terms by adding their coefficients:

  1. Identify like terms: In this case, the like terms are 12a and 4a.
  2. Combine the coefficients: Add the coefficients of the like terms: 12 + 4 = 16.
  3. Rewrite the expression: Replace the like terms with their combined form: 16a.

After combining like terms, the expression 12a + 20 + 4a simplifies to 16a + 20. This simplified form is identical to the expression we obtained earlier using the distributive property, confirming that 12a + 20 + 4a is equivalent to 4(4a + 5). This choice highlights the importance of simplifying expressions by combining like terms to reveal their underlying equivalence.

D. 2(8a + 10)

This expression, 2(8a + 10), presents another opportunity to apply the distributive property. Just like with the original expression, we can distribute the constant outside the parentheses (2) to each term inside the parentheses:

  1. Multiply 2 by 8a: 2 * 8a = 16a
  2. Multiply 2 by 10: 2 * 10 = 20
  3. Combine the results: 16a + 20

Applying the distributive property transforms 2(8a + 10) into 16a + 20, which is the same simplified form we obtained from the original expression. This confirms that 2(8a + 10) is indeed equivalent to 4(4a + 5). This choice reinforces the versatility of the distributive property in identifying equivalent expressions, even when they appear structurally different at first glance.

E. 16a + 5 + 4

This expression, 16a + 5 + 4, involves constant terms that can be combined. We can simplify the expression by adding the constant terms 5 and 4:

  1. Identify constant terms: The constant terms are 5 and 4.
  2. Combine the constants: Add the constants: 5 + 4 = 9
  3. Rewrite the expression: Replace the constants with their sum: 16a + 9

After combining the constants, the expression 16a + 5 + 4 simplifies to 16a + 9. This simplified form differs from our target expression 16a + 20, as the constant terms are different (9 versus 20). Therefore, 16a + 5 + 4 is not equivalent to 4(4a + 5). This choice emphasizes the importance of simplifying expressions fully before comparing them for equivalence. Failing to combine like terms can lead to incorrect conclusions about whether expressions are equivalent.

Conclusion: Identifying the Equivalent Expressions

Through the application of the distributive property and the combination of like terms, we have successfully identified the expressions equivalent to 4(4a + 5). The equivalent expressions from the given choices are:

  • B. 16a + 20
  • C. 12a + 20 + 4a
  • D. 2(8a + 10)

These expressions, while appearing different on the surface, all simplify to the same algebraic form, 16a + 20, demonstrating their equivalence. This exercise underscores the importance of mastering fundamental algebraic techniques like the distributive property and combining like terms. These skills are essential for simplifying expressions, solving equations, and developing a deeper understanding of mathematical relationships. Furthermore, the process of evaluating each answer choice systematically highlights the need for careful attention to detail and a thorough understanding of algebraic principles when determining equivalence. By mastering these concepts, you'll be well-equipped to tackle a wide range of algebraic problems and confidently navigate the world of mathematical expressions.

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