Solve: 3 # 5 # 2 Using Custom Operation

by Alex Johnson 40 views

In the realm of mathematics, we often encounter novel operations that challenge our conventional understanding of arithmetic. Today, we're diving into a unique mathematical expression where the symbol '#' doesn't represent multiplication or addition, but a custom-defined operation. Our task is to solve for 3 # 5 # 2 given the rule: x # y # z = x^3 + 5y - z. This problem is a fantastic way to practice substitution and order of operations in a new context. It requires us to carefully identify which number corresponds to x, y, and z in our given expression and then apply the defined formula. Let's break down how to approach this, ensuring we don't get tripped up by the unconventional notation. The key here is pattern recognition and careful application of the given rule. When you see x # y # z, you can think of it as a function where x, y, and z are inputs. In our specific problem, 3 # 5 # 2, the number 3 takes the place of x, the number 5 takes the place of y, and the number 2 takes the place of z. It's like solving a puzzle where each symbol has a specific role.

Understanding the Custom Operation: x # y # z = x^3 + 5y - z

Before we substitute our values, let's thoroughly understand the operation itself. The rule x # y # z = x^3 + 5y - z tells us exactly what to do with the three numbers involved. First, the first number (x) is cubed. This means we multiply x by itself three times (x * x * x). Second, the second number (y) is multiplied by 5. Finally, we subtract the third number (z) from the result of the first two operations. It's crucial to follow this order: cubing x, multiplying y by 5, and then subtracting z. The structure of the operation is clear: (x^3) + (5 * y) - z. By dissecting the formula like this, we can better prepare ourselves for the substitution process. This operation combines exponentiation, multiplication, and subtraction, showcasing how different mathematical concepts can be integrated into a single, custom rule. It's a great exercise to solidify your understanding of how mathematical functions work and how they operate on specific inputs. The more you practice with these custom operations, the more comfortable you'll become with abstract mathematical concepts and the flexibility of mathematical notation. Remember, mathematics is not just about memorizing formulas, but understanding the logic behind them and how they can be applied in various scenarios. This custom operation is a prime example of that.

Substituting Values into the Operation

Now comes the exciting part: putting our numbers into the formula! We have the expression 3 # 5 # 2 and the rule x # y # z = x^3 + 5y - z. Based on the order in our expression, we can confidently say that:

  • x = 3
  • y = 5
  • z = 2

Let's substitute these values directly into the formula x^3 + 5y - z:

3^3 + (5 * 5) - 2

See how straightforward it is? We've replaced x with 3, y with 5, and z with 2. The structure of the operation remains the same, but now it's filled with our specific numbers. This substitution is the core of solving problems involving custom operations. It requires close attention to detail to ensure each number goes into its correct place. Misplacing even one number could lead to an entirely different, incorrect answer. This is why it's always a good idea to double-check your substitutions before moving on to the calculation phase. Think of it like assigning roles in a play; each actor (number) must go to their designated spot on the stage (the formula). The precision in this step directly impacts the accuracy of your final result. Clarity in notation and careful mapping of variables are paramount. This methodical approach ensures that we are not just guessing, but applying a logical and systematic process to arrive at the solution. It’s the foundational step that enables all subsequent calculations to be accurate and meaningful within the context of the given problem.

Performing the Calculation: Step-by-Step

With our values substituted, we now need to perform the arithmetic operations according to the standard order of operations (often remembered by the acronym PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division from left to right, Addition and Subtraction from left to right). Our substituted expression is 3^3 + (5 * 5) - 2.

  1. Exponents: First, we tackle the exponent. 3^3 means 3 * 3 * 3. This equals 27.
  2. Multiplication: Next, we perform the multiplication. (5 * 5) equals 25.
  3. Addition and Subtraction: Now our expression looks like 27 + 25 - 2. We perform addition and subtraction from left to right.
    • 27 + 25 = 52
    • 52 - 2 = 50

So, 3 # 5 # 2 = 50.

This step-by-step calculation is crucial for accuracy. By breaking down the problem into smaller, manageable parts, we reduce the chance of making errors. We followed the defined operation rule precisely: we cubed the first number, multiplied the second by five, and then subtracted the third. The process highlights the importance of order of operations in mathematics, even when dealing with non-standard symbols. Each step builds upon the previous one, leading us logically to the final answer. Attention to detail at each stage ensures that the final result is reliable. This methodical breakdown not only helps in solving the current problem but also reinforces the fundamental arithmetic skills needed for more complex mathematical challenges. It’s a testament to how systematic approaches yield consistent and correct outcomes in mathematical problem-solving.

Conclusion: The Result of 3 # 5 # 2

After carefully applying the custom operation x # y # z = x^3 + 5y - z to the specific numbers 3 # 5 # 2, we arrive at our final answer. By substituting x=3, y=5, and z=2 into the formula, we performed the necessary calculations:

3^3 + (5 * 5) - 2

This evaluated to:

27 + 25 - 2

Which ultimately resulted in:

50

Therefore, the solution to 3 # 5 # 2 under the given custom operation is 50. This exercise demonstrates the power of defining new mathematical operations and the importance of precise substitution and calculation. It's a fun way to engage with mathematical concepts and practice fundamental skills. Problems like these encourage a flexible and analytical mindset, essential for tackling a wide range of mathematical challenges. They remind us that the symbolic language of mathematics can be extended and adapted, opening up new possibilities for problem-solving and discovery.

For those interested in exploring more about mathematical operations and functions, the Khan Academy website offers a vast array of resources, from basic arithmetic to advanced algebra, that can help deepen your understanding.