Step-by-Step Solution To 2^2 × 6 4/5 + 7 5/7

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In this comprehensive guide, we will delve into the step-by-step solution of the mathematical expression 2^2 × 6 4/5 + 7 5/7. This seemingly complex equation can be easily tackled by breaking it down into smaller, manageable steps. Our aim is to provide a clear and concise explanation that will empower you to solve similar problems with confidence. Whether you're a student looking to improve your math skills or simply someone who enjoys the challenge of mathematical problem-solving, this guide is designed to help you master the fundamentals of arithmetic operations and fractions.

Understanding the Order of Operations

Before we dive into the specifics of our equation, it's crucial to understand the order of operations. In mathematics, we follow a specific hierarchy to ensure that we arrive at the correct answer. This hierarchy is often remembered by the acronym PEMDAS, which stands for:

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

By adhering to PEMDAS, we can avoid ambiguity and ensure that our calculations are accurate. In our equation, 2^2 × 6 4/5 + 7 5/7, we'll first address the exponent, then the multiplication, and finally the addition.

Step 1: Solving the Exponent

The first step in solving 2^2 × 6 4/5 + 7 5/7 is to address the exponent. 2^2 means 2 raised to the power of 2, which is simply 2 multiplied by itself: 2 × 2 = 4. This calculation simplifies our equation to 4 × 6 4/5 + 7 5/7. Understanding exponents is crucial for many mathematical concepts, and this initial step lays the foundation for the rest of the solution.

Step 2: Converting Mixed Numbers to Improper Fractions

Next, we need to convert the mixed numbers, 6 4/5 and 7 5/7, into improper fractions. This conversion is essential for performing multiplication and addition with fractions. To convert a mixed number to an improper fraction, we multiply the whole number by the denominator and add the numerator. The result becomes the new numerator, and the denominator remains the same.

For 6 4/5, we calculate (6 × 5) + 4 = 30 + 4 = 34. So, 6 4/5 becomes 34/5.

Similarly, for 7 5/7, we calculate (7 × 7) + 5 = 49 + 5 = 54. Thus, 7 5/7 becomes 54/7.

Now our equation looks like this: 4 × 34/5 + 54/7. Converting mixed numbers to improper fractions is a fundamental skill in arithmetic, allowing us to perform operations more easily.

Step 3: Performing the Multiplication

Now that we have converted the mixed numbers to improper fractions, we can perform the multiplication. We need to multiply 4 by 34/5. To do this, we can treat 4 as a fraction, 4/1, and then multiply the numerators and the denominators:

(4/1) × (34/5) = (4 × 34) / (1 × 5) = 136/5

So, the equation now becomes 136/5 + 54/7. Multiplication of fractions is a straightforward process once the numbers are in the correct format, and this step brings us closer to the final solution.

Step 4: Finding a Common Denominator

Before we can add the fractions 136/5 and 54/7, we need to find a common denominator. The common denominator is the least common multiple (LCM) of the denominators, 5 and 7. Since 5 and 7 are both prime numbers, their LCM is simply their product: 5 × 7 = 35. Therefore, we need to convert both fractions to have a denominator of 35.

To convert 136/5 to a fraction with a denominator of 35, we multiply both the numerator and the denominator by 7:

(136/5) × (7/7) = (136 × 7) / (5 × 7) = 952/35

To convert 54/7 to a fraction with a denominator of 35, we multiply both the numerator and the denominator by 5:

(54/7) × (5/5) = (54 × 5) / (7 × 5) = 270/35

Now our equation looks like this: 952/35 + 270/35. Finding a common denominator is a crucial step in adding or subtracting fractions, ensuring that we are adding like terms.

Step 5: Adding the Fractions

With the fractions now having a common denominator, we can add them together. To add fractions with the same denominator, we simply add the numerators and keep the denominator the same:

952/35 + 270/35 = (952 + 270) / 35 = 1222/35

So, the sum of the fractions is 1222/35. Adding fractions becomes simple once they share a common denominator, and this step brings us very close to the final answer.

Step 6: Converting the Improper Fraction to a Mixed Number

The final step is to convert the improper fraction 1222/35 back to a mixed number. To do this, we divide the numerator by the denominator and express the remainder as a fraction. Dividing 1222 by 35 gives us a quotient of 34 and a remainder of 32.

1222 ÷ 35 = 34 with a remainder of 32

Therefore, 1222/35 is equivalent to the mixed number 34 32/35. This conversion provides a more intuitive understanding of the final result.

Final Answer and Conclusion

After meticulously following each step, we have arrived at the solution to the equation 2^2 × 6 4/5 + 7 5/7. The final answer is 34 32/35. This comprehensive guide has demonstrated how to break down a complex mathematical expression into smaller, manageable parts. By understanding the order of operations, converting mixed numbers to improper fractions, finding common denominators, and performing the necessary arithmetic operations, we have successfully solved the problem.

Key Takeaways

  • Order of Operations (PEMDAS): Always follow the correct order of operations to ensure accurate calculations.
  • Converting Mixed Numbers: Convert mixed numbers to improper fractions before performing multiplication or addition.
  • Finding Common Denominators: A common denominator is essential for adding and subtracting fractions.
  • Step-by-Step Approach: Breaking down complex problems into smaller steps makes them easier to solve.

By mastering these fundamental concepts and techniques, you can confidently tackle a wide range of mathematical problems. We hope this step-by-step guide has been helpful in enhancing your understanding of arithmetic operations and fractions. Whether you're a student or simply a math enthusiast, remember that practice and a methodical approach are key to success in mathematics. With this guide, you are now well-equipped to solve similar mathematical expressions with ease and precision.

In conclusion, solving mathematical problems like 2^2 × 6 4/5 + 7 5/7 is not just about arriving at the correct answer; it's about understanding the process and developing a systematic approach. Each step, from addressing the exponent to converting fractions and finding common denominators, plays a crucial role in the overall solution. By practicing these steps and applying the principles of PEMDAS, you can build a strong foundation in arithmetic and enhance your problem-solving skills. Remember, mathematics is a journey of learning and discovery, and every problem solved is a step forward in mastering this fascinating subject.