Solving Complex Math Expressions: A Step-by-Step Guide

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Solving Complex Math Expressions: A Step-by-Step Guide

Hey guys! Math can seem daunting, especially when you're faced with complex expressions. But don't worry, we're here to break it down step-by-step. In this article, we'll tackle some intricate mathematical problems and show you how to solve them with ease. So, grab your calculators and let's dive in!

a) (25:6 - 1300:13) : 10 - 676:169

When faced with a complex math expression, it's crucial to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This ensures we solve the expression in the correct sequence, leading to the accurate answer. Let's apply this principle to the first expression.

Breaking Down the Expression

The expression is: (25:6 - 1300:13) : 10 - 676:169. We'll tackle it piece by piece.

  1. Parentheses First: Inside the parentheses, we have division operations. So, we'll start there.
    • 25 divided by 6 can be written as 25/6 which equals approximately 4.1667.
    • Next, 1300 divided by 13 is 100. Simple enough, right?
  2. Subtraction Inside Parentheses: Now we subtract the results within the parentheses.
    • So, it's roughly 4.1667 - 100, which gives us -95.8333.
  3. Division: We're moving outside the parentheses now. We have two division operations to handle.
    • First, divide the result from the parentheses (-95.8333) by 10, resulting in -9.5833.
    • Then, divide 676 by 169, which equals 4.
  4. Final Subtraction: Lastly, subtract the results of the division operations.
    • This means -9.5833 - 4, which gives us the final result of -13.5833.

Therefore, the solution to the expression (25:6 - 1300:13) : 10 - 676:169 is approximately -13.5833. Remember, guys, the key is to take it one step at a time, following the order of operations.

b) 416:4 - 5 โ‹… (361:19 - 256:32)

Now, let's move on to the second expression. This one involves multiple operations, including multiplication and parentheses. Again, the order of operations (PEMDAS) is our best friend here. We'll break down the expression step by step, making sure to handle each part carefully.

Step-by-Step Solution

The expression we're tackling is: 416:4 - 5 โ‹… (361:19 - 256:32).

  1. Parentheses First: Just like before, we start inside the parentheses.
    • We have two division operations here: 361 divided by 19 and 256 divided by 32.
    • 361 divided by 19 equals 19.
    • 256 divided by 32 equals 8.
  2. Subtraction Inside Parentheses: Now we subtract the results within the parentheses.
    • So, it's 19 - 8, which equals 11. Easy peasy, right?
  3. Multiplication: Next, we handle the multiplication operation.
    • We need to multiply 5 by the result we got from the parentheses, which is 11.
    • 5 multiplied by 11 equals 55.
  4. Division: Now we move to the division part of the expression.
    • 416 divided by 4 equals 104.
  5. Final Subtraction: Finally, we perform the subtraction operation.
    • We subtract the result of the multiplication (55) from the result of the division (104).
    • So, 104 - 55 gives us the final answer.

Therefore, the solution to the expression 416:4 - 5 โ‹… (361:19 - 256:32) is 49. Keep practicing, and these steps will become second nature to you!

c) 720:90 + (2 โ‹… 19 - 2 โ‹… 16) โ‹… 12 - 36:6

Moving on to our third expression, we're faced with a combination of division, multiplication, addition, and subtraction. The key, as always, is to stick to the order of operations (PEMDAS). This ensures we tackle each part in the correct sequence, leading to an accurate solution. Let's break it down together.

Solving the Expression Step-by-Step

Here's the expression we're working with: 720:90 + (2 โ‹… 19 - 2 โ‹… 16) โ‹… 12 - 36:6.

  1. Parentheses First: We kick things off by focusing on the operations within the parentheses.
    • Inside, we have two multiplication operations: 2 multiplied by 19 and 2 multiplied by 16.
    • 2 times 19 equals 38.
    • 2 times 16 equals 32.
  2. Subtraction Inside Parentheses: Now, we subtract the results within the parentheses.
    • So, 38 - 32 equals 6.
  3. Multiplication: Moving outside the parentheses, we encounter a multiplication operation.
    • We need to multiply the result from the parentheses (6) by 12.
    • 6 multiplied by 12 equals 72.
  4. Division: Next up, we have two division operations.
    • First, 720 divided by 90 equals 8.
    • Then, 36 divided by 6 equals 6.
  5. Addition and Subtraction: We're in the final stretch! Now we perform the addition and subtraction from left to right.
    • First, add 8 (the result of 720:90) to 72 (the result of the multiplication), which gives us 80.
    • Then, subtract 6 (the result of 36:6) from 80.

Therefore, the solution to the expression 720:90 + (2 โ‹… 19 - 2 โ‹… 16) โ‹… 12 - 36:6 is 74. Remember, practice makes perfect, so keep at it!

d) [35 โ‹… 3 - (67 + 33)] โ‹… 10

This expression introduces brackets, but don't let that intimidate you! Brackets are just another set of parentheses, so we treat them the same way in the order of operations (PEMDAS). We'll start inside the brackets and work our way out, step by careful step.

Solving the Expression

Our expression is: [35 โ‹… 3 - (67 + 33)] โ‹… 10.

  1. Innermost Parentheses First: We begin with the innermost set of parentheses.
    • Here, we have an addition operation: 67 plus 33.
    • 67 plus 33 equals 100.
  2. Brackets: Now we move to the brackets. Treat the brackets just like regular parentheses.
    • Inside the brackets, we have a multiplication and a subtraction.
    • Following the order of operations, we do the multiplication first: 35 multiplied by 3 equals 105.
    • Next, we subtract the result from the innermost parentheses (100) from the result of the multiplication (105).
    • So, 105 - 100 equals 5.
  3. Multiplication: Finally, we have a multiplication operation outside the brackets.
    • We multiply the result from the brackets (5) by 10.

Therefore, the solution to the expression [35 โ‹… 3 - (67 + 33)] โ‹… 10 is 50. See? Not so scary when you break it down!

e) 1 + 2 + 3 + 4 โ‹… [5 + 6 โ‹… (216:36 - 54:9)]

This expression might look a bit intimidating with nested parentheses and multiple operations, but don't sweat it! We'll tackle it using our trusty friend, the order of operations (PEMDAS). The key here is to work from the innermost parentheses outwards, step by step.

Step-by-Step Solution

Let's break down the expression: 1 + 2 + 3 + 4 โ‹… [5 + 6 โ‹… (216:36 - 54:9)].

  1. Innermost Parentheses: We start with the innermost parentheses.
    • Inside, we have two division operations: 216 divided by 36 and 54 divided by 9.
    • 216 divided by 36 equals 6.
    • 54 divided by 9 equals 6.
  2. Subtraction Inside Parentheses: Now, we subtract the results within the parentheses.
    • So, 6 - 6 equals 0. That simplifies things nicely!
  3. Brackets: Moving to the brackets, we have a multiplication and an addition.
    • Following the order of operations, we do the multiplication first: 6 multiplied by the result from the parentheses (0) equals 0.
    • Next, we add 5 to the result of the multiplication (0).
    • So, 5 + 0 equals 5.
  4. Multiplication: Now we have a multiplication operation outside the brackets.
    • We multiply 4 by the result from the brackets (5).
    • 4 multiplied by 5 equals 20.
  5. Addition: Finally, we perform the addition operations from left to right.
    • 1 + 2 + 3 + 20. Let's do it step by step: 1 + 2 = 3, then 3 + 3 = 6, and finally, 6 + 20 = 26.

Therefore, the solution to the expression 1 + 2 + 3 + 4 โ‹… [5 + 6 โ‹… (216:36 - 54:9)] is 26. You got this, guys!

f) [(319 + 171):7 โ‹… 2 + 5 โ‹… (77 - 65)]:(7 + 93)

This expression looks long, but don't worry, we'll tackle it just like the others โ€“ step by step, following the order of operations (PEMDAS). We'll work from the innermost parentheses outwards, making sure to handle each operation carefully.

Solving the Complex Expression

Here's the expression we're going to solve: [(319 + 171):7 โ‹… 2 + 5 โ‹… (77 - 65)]:(7 + 93).

  1. Innermost Parentheses: Let's start with the innermost parentheses.
    • We have an addition within the first set of parentheses: 319 + 171.
    • 319 plus 171 equals 490.
    • In the other set of parentheses, we have a subtraction: 77 - 65.
    • 77 minus 65 equals 12.
  2. Brackets: Now, let's move to the brackets. We'll treat them as regular parentheses.
    • Inside the first set of brackets, we have division, multiplication, and addition. Following the order of operations, we do the division first: 490 divided by 7.
    • 490 divided by 7 equals 70.
    • Next, we multiply the result of the division (70) by 2.
    • 70 multiplied by 2 equals 140.
    • In the second part of the brackets, we have a multiplication: 5 multiplied by the result from the innermost parentheses (12).
    • 5 multiplied by 12 equals 60.
    • Now, we add the results within the first set of brackets: 140 + 60.
    • 140 plus 60 equals 200.
  3. Outermost Parentheses: Now, let's move to the outermost parentheses.
    • We have an addition operation: 7 + 93.
    • 7 plus 93 equals 100.
  4. Final Division: Finally, we have a division operation.
    • We divide the result from the brackets (200) by the result from the outermost parentheses (100).

Therefore, the solution to the expression [(319 + 171):7 โ‹… 2 + 5 โ‹… (77 - 65)]:(7 + 93) is 2. You're doing great, guys!

g) [214 โ‹… 84 - 3 โ‹… (10 - 32)] โ‹… [(22)3]

This expression involves exponents, but don't let that throw you off! We'll still rely on our trusty order of operations (PEMDAS). Remember that exponents come before multiplication and division. Let's break it down step by step.

Solving the Expression with Exponents

Here's the expression we're tackling: [214 โ‹… 84 - 3 โ‹… (10 - 32)] โ‹… [(22)3].

  1. Parentheses First: Let's start with the parentheses.
    • Inside the parentheses, we have a subtraction: 10 - 32.
    • 10 minus 32 equals -22. Don't forget the negative sign!
  2. Brackets: Now we move to the brackets. We'll treat them like regular parentheses.
    • In the first set of brackets, we have multiplication and subtraction. Following the order of operations, we do the multiplication first: 3 multiplied by the result from the parentheses (-22).
    • 3 multiplied by -22 equals -66.
    • Next, we have another multiplication: 214 multiplied by 84.
    • 214 multiplied by 84 equals 17976.
    • Now, we subtract the result of the first multiplication (-66) from the result of the second multiplication (17976). Remember that subtracting a negative is the same as adding, so it's 17976 - (-66), which is the same as 17976 + 66.
    • 17976 plus 66 equals 18042.
    • In the second set of brackets, we have exponents. We start with the innermost exponent: 2 squared (2^2).
    • 2 squared (2^2) equals 4.
    • Now we raise the result (4) to the power of 3: 4 cubed (4^3).
    • 4 cubed (4^3) equals 64.
  3. Final Multiplication: Finally, we have a multiplication operation between the results from the two sets of brackets.
    • We multiply the result from the first set of brackets (18042) by the result from the second set of brackets (64).

Therefore, the solution to the expression [214 โ‹… 84 - 3 โ‹… (10 - 32)] โ‹… [(22)3] is 1154688. You're becoming math wizards, guys!

Mastering Math Expressions: Key Takeaways

We've tackled some pretty complex math expressions in this guide, and you've seen how to break them down step by step. Remember these key principles to conquer any mathematical challenge:

  • Order of Operations (PEMDAS): This is your best friend! Always follow the order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
  • Break It Down: Complex expressions can seem overwhelming, but break them down into smaller, manageable parts. Focus on one operation at a time.
  • Work from the Inside Out: When dealing with nested parentheses or brackets, start with the innermost set and work your way outwards.
  • Practice Makes Perfect: The more you practice, the more comfortable you'll become with these concepts. Don't be afraid to make mistakes โ€“ they're part of the learning process!

So there you have it, guys! You're now equipped to handle even the trickiest math expressions. Keep practicing, stay confident, and you'll be solving problems like a pro in no time!