Math Problems: Step-by-Step Solutions And Explanations

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Math Problems: Step-by-Step Solutions and Explanations

Hey everyone! Today, we're diving into some math problems. Don't worry, we'll break down each one step by step, so even if you're not a math whiz, you'll be able to follow along. We'll cover everything from basic arithmetic to square roots and fractions. Let's get started, shall we?

Solving Arithmetic and Fraction Problems

Let's tackle the first part of the problem. This involves a mix of addition, subtraction, multiplication, and fractions. Remember to follow the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This ensures we solve the problem in the correct sequence, avoiding any potential mix-ups. This is super important, guys.

Problem Breakdown

The first problem is:

12 + 24 + 54 + 15 - (2 * 5 * 2/12) + 24 + 11/6

  1. Addition: Start by adding the whole numbers: 12 + 24 + 54 + 15 + 24 = 129.
  2. Parentheses: Next, solve the expression inside the parentheses: 2 * 5 * (2/12). Simplify the fraction 2/12 to 1/6. Now, the expression becomes 2 * 5 * (1/6) = 10/6, which further simplifies to 5/3.
  3. Subtraction: Subtract the result from the parentheses: 129 - 5/3. To do this, convert 129 into a fraction with a denominator of 3: 129 = 387/3. Thus, 387/3 - 5/3 = 382/3.
  4. Addition of Fractions: Now, deal with the remaining fraction: 11/6. To add this to 382/3, we must have a common denominator. Multiplying the numerator and denominator of 382/3 by 2 yields 764/6. Finally, add the fractions 764/6 + 11/6 = 775/6.

So, the answer for the first part is 775/6, which is approximately 129.1667. See, not so scary, right? Remember to always break down problems into smaller, manageable steps. This not only makes them easier to solve but also helps to minimize errors.

Practical Tips for Accuracy

  • Double-check: Always review each calculation. Simple errors, like misreading a number, can significantly impact your final answer. Taking a couple of extra seconds to check your work can save you a lot of headache.
  • Use a calculator: While it's great to practice mental math, don't hesitate to use a calculator for more complex calculations, especially when dealing with decimals or fractions. This reduces the risk of making arithmetic errors. However, make sure you understand the underlying mathematical principles before relying solely on a calculator.
  • Write clearly: Neatness counts! Keeping your work organized makes it easier to track your steps and spot mistakes. If your writing is messy, it will be difficult to review your work.

Tackling Problems with Square Roots and More

Now, let's explore the second part of the problem, which includes square roots and more mixed operations. This is a great chance to enhance your understanding of radicals and their properties. We will apply the order of operations and learn a few tricks on how to simplify those square roots.

Problem Breakdown

The second problem is:

15 + 27 + 12 + 12 + 20 + 54 * (√24 / √96) + 18 + 7 + 9 + 27 * (10/34)

  1. Addition: Add the whole numbers: 15 + 27 + 12 + 12 + 20 + 18 + 7 + 9 = 120.
  2. Square Roots: Simplify the square root expression: √(24) / √(96). Recall that √24 can be simplified to 2√6 and √96 can be simplified to 4√6. Therefore, √(24) / √(96) is equivalent to 2√6 / 4√6. By canceling the √6 terms, we have 2/4 = 1/2.
  3. Multiplication: Multiply: 54 * (1/2) = 27.
  4. Fractions: Simplify the fraction 10/34 to 5/17. Multiply: 27 * (5/17) = 135/17. This simplifies to approximately 7.94. This step involves simplifying the original fractions and performing the multiplication.
  5. Final Addition: Add all the results: 120 + 27 + 135/17 = 147 + 135/17 = 262/17 which is about 174.94. Pretty cool, eh? This gives the final answer for the second part of the problem.

Square Root Simplification

Simplifying square roots involves finding perfect square factors within the number under the radical. For example, √24 can be written as √(4 * 6), which simplifies to 2√6 because √4 = 2. Recognizing perfect squares allows you to reduce the radical to its simplest form. It's really about knowing the times tables and recognizing when a number can be broken down.

  • Perfect Squares: Memorize perfect squares: 4, 9, 16, 25, 36, 49, 64, 81, 100, etc. Knowing these helps you quickly identify and extract perfect square factors. Knowing the perfect square will make it easier.
  • Prime Factorization: If you find it difficult to identify perfect square factors, use prime factorization to break down the number into its prime components. This method is particularly helpful with larger numbers.

Simplifying Square Root Expressions

Now, let's tackle an expression solely focused on square roots, showcasing how to combine them effectively. Remember that we can only add or subtract square roots if the expressions under the radical are the same. Let's delve in and find out what to do.

Problem Breakdown

The third problem is:

√3 + 5√3 + 10√3 + 4√3 + 16 + 18

  1. Addition of like terms: Add the terms with √3: √3 + 5√3 + 10√3 + 4√3 = 20√3.
  2. Addition of whole numbers: Add the whole numbers: 16 + 18 = 34.
  3. Final Answer: Combine the results: 20√3 + 34. Since √3 is an irrational number, you cannot further simplify this expression. The final answer is 20√3 + 34, which is approximately 68.64. Great work, you are really getting the hang of this!

Simplifying Square Root Expressions Tips

  • Identify Like Radicals: First, ensure that the numbers under the square root symbol (radicands) are identical before attempting addition or subtraction. You cannot directly add or subtract √2 and √3. Ensure that you have like terms to simplify the expression.
  • Simplify Radicals if Needed: If your expressions have different numbers inside the square root, try simplifying them first. This could lead to like terms that you can add or subtract. This is important to ensure that you are able to perform the simplification.
  • Factor out Perfect Squares: Before adding or subtracting, simplify any square roots by factoring out perfect squares. For instance, √12 is the same as √(4 * 3), which simplifies to 2√3.

Combining Fractions and Square Roots

Finally, let's combine all of our skills and knowledge to the final part of the problem, which combines fractions and square roots. Let's conquer the final frontier of this math problem together.

Problem Breakdown

The fourth problem is:

(15/6) + 5√6 + √54 + √24 + 3/54

  1. Fractions: Simplify the fraction 15/6 to 5/2. The fraction 3/54 can be simplified to 1/18. Therefore, the fractions becomes 5/2 + 1/18.
  2. Square Roots: Simplify √54 to 3√6 and √24 to 2√6.
  3. Combine Like Terms: Group all the terms with √6 and add: 5√6 + 3√6 + 2√6 = 10√6.
  4. Finding a Common Denominator: Add the fractions: 5/2 + 1/18. The common denominator is 18. Rewrite 5/2 as 45/18. Hence, 45/18 + 1/18 = 46/18.
  5. Final Answer: Combine the results: 10√6 + 46/18. Simplifying the fraction 46/18 results in 23/9. The answer becomes 10√6 + 23/9. This is approximately 29.80.

Practical Application

Combining fractions and square roots requires careful attention to detail. Always prioritize simplifying your fractions and square roots before combining them. Simplify fractions to their lowest terms and reduce the square roots to a more simplified form. This will make your calculations easier and will reduce the chance of making errors. Awesome, you made it all the way through! Using calculators can be helpful at this stage, but make sure that you know the underlying principles behind the answer.

Keep Practicing!

The key to improving your math skills is practice! Keep working on similar problems, and don't be afraid to ask for help when you need it. You got this, guys! With each problem, you'll become more confident and proficient.