Math Operations: Calculate The Results Step-by-Step

Hey there, math enthusiasts! Today, we're diving into the nitty-gritty of calculating the results of several math operations. Whether you're brushing up on your skills or tackling a tricky problem, this guide will walk you through each step. So grab your calculators (or your mental math muscles) and let's get started!

1. Understanding the Basics

Before we jump into the calculations, let's quickly recap the order of operations. Remember PEMDAS/BODMAS:

• Parentheses / Brackets
• Exponents / Orders
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

This order is super crucial for getting the correct answers. Mess it up, and you might end up with a totally different result. So keep PEMDAS/BODMAS in mind as we go through each problem. Understanding this order is the first step to mastering these calculations.

2. Problem 1: $10 + 26 - 31 - 12 + 8 - 9$

Alright, let's kick things off with a relatively simple one. This problem involves only addition and subtraction, so we'll work from left to right.

• Start with 10 + 26, which equals 36.
• Then, 36 - 31 gives us 5.
• Next, 5 - 12 results in -7.
• After that, -7 + 8 equals 1.
• Finally, 1 - 9 leaves us with -8.

So, the result of $10 + 26 - 31 - 12 + 8 - 9$ is -8. See? Not too scary when you break it down step by step. Breaking down the problem into smaller steps is the key to solving it easily.

3. Problem 2: $-15 + 13 + 20 - 42 + 15$

Moving on, let's tackle another addition and subtraction problem. Again, we'll work from left to right.

• First, -15 + 13 equals -2.
• Then, -2 + 20 gives us 18.
• Next, 18 - 42 results in -24.
• Finally, -24 + 15 leaves us with -9.

Therefore, the result of $-15 + 13 + 20 - 42 + 15$ is -9. Notice how handling negative numbers is just as straightforward as positive ones, as long as you keep track of the signs. Managing negative numbers correctly is vital for accurate calculations.

4. Problem 3: $4 imes 9 + 6

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÷ 3 imes 7 - 6 + 4$

Okay, things are getting a little more interesting now! This problem includes multiplication, division, addition, and subtraction. Remember PEMDAS/BODMAS? Multiplication and division come before addition and subtraction, and we perform them from left to right.

• First, 4 × 9 equals 36.
• Next, 6 ÷ 3 equals 2.
• Then, 2 × 7 equals 14.
• Now we have: 36 + 14 - 6 + 4
• Next, 36 + 14 equals 50.
• Then, 50 - 6 equals 44.
• Finally, 44 + 4 gives us 48.

So, the result of $4 \times 9 + 6 \div 3 \times 7 - 6 + 4$ is 48. This problem highlights the importance of following the order of operations. Following the order of operations ensures the correct outcome.

5. Problem 4: $40

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÷ 2^3 - 7 imes 4 + 6^2 ewline

÷ 4 - 8 imes 3$

This one's got exponents in the mix! Don't worry, we've got this. Exponents come before multiplication and division in PEMDAS/BODMAS.

• First, calculate the exponents: 2^3 equals 8, and 6^2 equals 36.
• Now we have: 40 ÷ 8 - 7 × 4 + 36 ÷ 4 - 8 × 3
• Next, perform the division and multiplication from left to right:
  • 40 ÷ 8 equals 5
  • 7 × 4 equals 28
  • 36 ÷ 4 equals 9
  • 8 × 3 equals 24
• Now we have: 5 - 28 + 9 - 24
• Next, perform the addition and subtraction from left to right:
  • 5 - 28 equals -23
  • -23 + 9 equals -14
  • -14 - 24 equals -38

Therefore, the result of $40 \div 2^3 - 7 \times 4 + 6^2 \div 4 - 8 \times 3$ is -38. Exponents might seem intimidating, but they're just another step in the process. Mastering exponents is crucial for complex calculations.

6. Problem 5: $-7 + \{(-4 + 3) - 8 - (2 + 5)^2\} imes 2$

Brace yourselves, guys! This one's got brackets, parentheses, and exponents. But don't sweat it – we'll take it one layer at a time, starting with the innermost parentheses.

• First, work inside the innermost parentheses: -4 + 3 equals -1 and 2 + 5 equals 7.
• Now we have: -7 + {-1 - 8 - 7^2} × 2
• Next, calculate the exponent: 7^2 equals 49.
• Now we have: -7 + {-1 - 8 - 49} × 2
• Then, work inside the curly brackets: -1 - 8 - 49 equals -58.
• Now we have: -7 + (-58) × 2
• Next, perform the multiplication: -58 × 2 equals -116.
• Now we have: -7 + (-116)
• Finally, -7 + (-116) equals -123.

So, the result of $-7 + \{(-4 + 3) - 8 - (2 + 5)^2\} \times 2$ is -123. This problem is a fantastic example of how breaking down complex expressions into smaller parts makes them much more manageable. Breaking down expressions simplifies the calculation process.

7. Problem 6: $12 + [-5 + \{-3 + 8(4 + 6)\} \div 5]$

Last but not least, let's tackle this beast of a problem! We've got nested brackets and parentheses, but we'll conquer it just like the others – step by step.

• First, work inside the innermost parentheses: 4 + 6 equals 10.
• Now we have: 12 + [-5 + {-3 + 8(10)} ÷ 5]
• Next, perform the multiplication inside the curly brackets: 8(10) equals 80.
• Now we have: 12 + [-5 + {-3 + 80} ÷ 5]
• Then, continue working inside the curly brackets: -3 + 80 equals 77.
• Now we have: 12 + [-5 + 77 ÷ 5]
• Next, perform the division: 77 ÷ 5 equals 15.4.
• Now we have: 12 + [-5 + 15.4]
• Then, work inside the square brackets: -5 + 15.4 equals 10.4.
• Now we have: 12 + 10.4
• Finally, 12 + 10.4 equals 22.4.

Therefore, the result of $12 + [-5 + \{-3 + 8(4 + 6)\} \div 5]$ is 22.4. This problem really puts our order of operations knowledge to the test, and we nailed it! Testing your knowledge with complex problems is a great way to improve.

Conclusion

Alright, guys, that's a wrap! We've tackled some pretty intense math problems today, and you've seen how breaking them down step by step, following PEMDAS/BODMAS, can make even the trickiest calculations manageable. Keep practicing, and you'll be a math whiz in no time! Remember, practice makes perfect, so keep those numbers crunching!