Equivalent Of 9² + 4² Exploring Mathematical Expressions

by ADMIN 57 views

In the realm of mathematics, understanding fundamental operations and their properties is crucial for problem-solving and building a strong foundation. One such concept involves exploring the equivalence of mathematical expressions. This article delves into the expression $9^2 + 4^2$, dissecting its components and comparing it with various alternatives to determine the correct equivalence. Our primary focus revolves around identifying the expression that yields the same result as $9^2 + 4^2$, emphasizing the significance of order of operations and the distributive property.

Deconstructing the Expression: 9² + 4²

Before we embark on comparing the expression with the given options, let's break it down to its core elements. The expression $9^2 + 4^2$ involves two terms: $9^2$ and $4^2$. The superscript '2' indicates that each number is raised to the power of 2, which means it is multiplied by itself. Therefore, $9^2$ is equivalent to 9 multiplied by 9 (9 * 9), and $4^2$ is equivalent to 4 multiplied by 4 (4 * 4). To find the value of the entire expression, we need to calculate the values of $9^2$ and $4^2$ individually and then add them together. Following the order of operations (PEMDAS/BODMAS), exponents are evaluated before addition. Thus, the calculation unfolds as follows:

  • 92=9imes9=819^2 = 9 imes 9 = 81

  • 42=4imes4=164^2 = 4 imes 4 = 16

Adding these results together:

81+16=9781 + 16 = 97

Therefore, the value of the expression $9^2 + 4^2$ is 97. This value will serve as our benchmark as we evaluate the provided options.

Evaluating Option A: (9 + 4)²

Option A presents the expression $(9 + 4)^2$, which may appear similar to the original expression but involves a crucial difference in the order of operations. In this case, the numbers 9 and 4 are first added together, and then the result is raised to the power of 2. To evaluate this expression, we follow these steps:

  1. Add 9 and 4: $9 + 4 = 13$
  2. Raise the result to the power of 2: $13^2 = 13 imes 13 = 169$

The value of the expression $(9 + 4)^2$ is 169. Comparing this with the value of the original expression (97), we can clearly see that they are not equivalent. This is because squaring the sum of two numbers is not the same as summing the squares of the individual numbers. This distinction highlights the importance of adhering to the order of operations and understanding the properties of exponents.

Assessing Option B: (9 × 9) + (4 × 4)

Option B offers the expression $(9 imes 9) + (4 imes 4)$. This expression explicitly shows the multiplication of each number by itself, which aligns with the definition of squaring a number. To evaluate this expression, we perform the multiplications first and then add the results:

  1. Multiply 9 by 9: $9 imes 9 = 81$
  2. Multiply 4 by 4: $4 imes 4 = 16$
  3. Add the results: $81 + 16 = 97$

The value of the expression $(9 imes 9) + (4 imes 4)$ is 97. This value is identical to the value of the original expression, $9^2 + 4^2$. Therefore, Option B is equivalent to the original expression. This equivalence arises because the expression explicitly expands the squaring operation, demonstrating the fundamental definition of raising a number to the power of 2.

Examining Option C: (9 + 9) + (4 + 4)

Option C presents the expression $(9 + 9) + (4 + 4)$. This expression involves adding each number to itself, which is equivalent to multiplying each number by 2. To evaluate this expression, we perform the additions within the parentheses and then add the results:

  1. Add 9 and 9: $9 + 9 = 18$
  2. Add 4 and 4: $4 + 4 = 8$
  3. Add the results: $18 + 8 = 26$

The value of the expression $(9 + 9) + (4 + 4)$ is 26. Comparing this with the value of the original expression (97), it is evident that they are not equivalent. This expression represents doubling each number and then summing the results, which is distinct from squaring each number and then summing the results.

Analyzing Option D: (9 × 4)²

Option D offers the expression $(9 imes 4)^2$. This expression involves multiplying 9 and 4 first, and then squaring the result. To evaluate this expression, we follow these steps:

  1. Multiply 9 and 4: $9 imes 4 = 36$
  2. Square the result: $36^2 = 36 imes 36 = 1296$

The value of the expression $(9 imes 4)^2$ is 1296. This value is significantly different from the value of the original expression (97), indicating that they are not equivalent. Squaring the product of two numbers is not the same as summing the squares of the individual numbers. This difference highlights the importance of understanding the distributive property and the order of operations.

Concluding the Equivalence

After meticulously evaluating each option, we have determined that Option B, (9 × 9) + (4 × 4), is the only expression equivalent to $9^2 + 4^2$. This equivalence stems from the direct application of the definition of squaring a number, where each number is multiplied by itself. The other options involved different operations or altered the order of operations, leading to results that did not match the original expression's value.

This exploration underscores the significance of a solid grasp of mathematical principles, including the order of operations and the properties of exponents. By carefully dissecting expressions and applying these principles, we can confidently determine equivalence and solve mathematical problems with accuracy. The expression $9^2 + 4^2$ serves as a valuable example to illustrate these concepts and reinforce the importance of precision in mathematical calculations.

In conclusion, when faced with expressions involving exponents and multiple operations, it is crucial to break down the components, apply the order of operations correctly, and compare the results to determine equivalence. This approach not only aids in solving specific problems but also strengthens the overall understanding of mathematical relationships and properties.