Evaluating Expressions A Step-by-Step Guide To Solving $\frac{3}{4}+3^3-\frac{1}{2} \div 2 \frac{2}{3}$

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Evaluating mathematical expressions requires a systematic approach, adhering to the order of operations and simplifying at each step. In this comprehensive guide, we will delve into the process of evaluating the expression 34+33−12÷223\frac{3}{4}+3^3-\frac{1}{2} \div 2 \frac{2}{3}, providing a clear and concise breakdown of each operation. Our aim is to express the final solution as a mixed number or a fraction in its lowest terms, ensuring a thorough understanding of the concepts involved.

Understanding the Order of Operations

Before we begin, it's crucial to grasp the fundamental principle that governs mathematical calculations: the order of operations. Often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), this rule dictates the sequence in which operations must be performed to arrive at the correct answer. Let's break down each component of PEMDAS:

  1. Parentheses: Operations within parentheses or brackets are always performed first. This ensures that expressions within these groupings are treated as a single entity.
  2. Exponents: Exponents, which represent repeated multiplication, are evaluated next. This step involves calculating the value of a number raised to a power.
  3. Multiplication and Division: Multiplication and division are performed from left to right. These operations have equal priority, so their order is determined by their position in the expression.
  4. Addition and Subtraction: Finally, addition and subtraction are carried out from left to right. Similar to multiplication and division, these operations have equal priority and are performed in the order they appear.

By adhering to PEMDAS, we can ensure that we evaluate expressions consistently and accurately. This foundation is essential for tackling more complex mathematical problems.

Step-by-Step Evaluation

Now that we have a firm understanding of the order of operations, let's apply it to the expression 34+33−12÷223\frac{3}{4}+3^3-\frac{1}{2} \div 2 \frac{2}{3}. We'll break down the evaluation into manageable steps, explaining each operation in detail.

1. Evaluate the Exponent

The first step in our evaluation is to address the exponent, 333^3. This represents 3 multiplied by itself three times:

33=3×3×3=273^3 = 3 \times 3 \times 3 = 27

Substituting this value back into the expression, we get:

34+27−12÷223\frac{3}{4}+27-\frac{1}{2} \div 2 \frac{2}{3}

2. Convert the Mixed Number to an Improper Fraction

Next, we need to convert the mixed number, 2232 \frac{2}{3}, into an improper fraction. To do this, we multiply the whole number (2) by the denominator (3) and add the numerator (2), then place the result over the original denominator:

223=(2×3)+23=832 \frac{2}{3} = \frac{(2 \times 3) + 2}{3} = \frac{8}{3}

Now our expression looks like this:

34+27−12÷83\frac{3}{4}+27-\frac{1}{2} \div \frac{8}{3}

3. Perform the Division

Following the order of operations, we perform the division next. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of 83\frac{8}{3} is 38\frac{3}{8}. Therefore,

12÷83=12×38=1×32×8=316\frac{1}{2} \div \frac{8}{3} = \frac{1}{2} \times \frac{3}{8} = \frac{1 \times 3}{2 \times 8} = \frac{3}{16}

Substituting this result back into the expression, we have:

34+27−316\frac{3}{4}+27-\frac{3}{16}

4. Find a Common Denominator

Before we can perform the addition and subtraction, we need to find a common denominator for the fractions 34\frac{3}{4} and 316\frac{3}{16}. The least common multiple (LCM) of 4 and 16 is 16. So, we convert 34\frac{3}{4} to an equivalent fraction with a denominator of 16:

34=3×44×4=1216\frac{3}{4} = \frac{3 \times 4}{4 \times 4} = \frac{12}{16}

Now our expression is:

1216+27−316\frac{12}{16}+27-\frac{3}{16}

5. Perform the Addition and Subtraction

Now we can perform the addition and subtraction from left to right. First, we add 1216\frac{12}{16} and 27. To do this, we can think of 27 as a fraction with a denominator of 1:

27=27127 = \frac{27}{1}

To add this to 1216\frac{12}{16}, we need a common denominator, which is 16. So, we convert 271\frac{27}{1} to an equivalent fraction with a denominator of 16:

271=27×161×16=43216\frac{27}{1} = \frac{27 \times 16}{1 \times 16} = \frac{432}{16}

Now we can add:

1216+43216=12+43216=44416\frac{12}{16} + \frac{432}{16} = \frac{12 + 432}{16} = \frac{444}{16}

Next, we subtract 316\frac{3}{16}:

44416−316=444−316=44116\frac{444}{16} - \frac{3}{16} = \frac{444 - 3}{16} = \frac{441}{16}

6. Express the Solution as a Mixed Number

Finally, we express the improper fraction 44116\frac{441}{16} as a mixed number. To do this, we divide 441 by 16:

441÷16=27441 \div 16 = 27 with a remainder of 9.

So, the mixed number is:

2791627 \frac{9}{16}

Therefore, the final answer is 2791627 \frac{9}{16}.

Key Concepts and Rules Used

Throughout the evaluation process, we employed several key mathematical concepts and rules. Let's recap these fundamentals to reinforce your understanding:

  • Order of Operations (PEMDAS): This rule ensures consistent and accurate evaluation of expressions.
  • Converting Mixed Numbers to Improper Fractions: This conversion is essential for performing operations with fractions.
  • Dividing by a Fraction: Dividing by a fraction is equivalent to multiplying by its reciprocal.
  • Finding a Common Denominator: A common denominator is necessary for adding and subtracting fractions.
  • Converting Improper Fractions to Mixed Numbers: This conversion allows us to express the solution in a more conventional form.

By mastering these concepts, you'll be well-equipped to tackle a wide range of mathematical expressions.

Alternative Approaches and Insights

While we followed a step-by-step approach to evaluate the expression, it's worth noting that there might be alternative routes to the solution. For instance, one could choose to combine the fractions earlier in the process. However, adhering to the order of operations generally provides a clear and organized path, minimizing the chances of errors.

Another valuable insight is the importance of checking your work. After each step, take a moment to review your calculations and ensure that you haven't made any mistakes. This practice can save you from arriving at an incorrect final answer.

Practice Problems

To solidify your understanding of evaluating expressions, try working through these practice problems:

  1. 12+24−34÷112\frac{1}{2} + 2^4 - \frac{3}{4} \div 1 \frac{1}{2}
  2. 52−23+14×3135^2 - \frac{2}{3} + \frac{1}{4} \times 3 \frac{1}{3}
  3. 58÷23+42−12\frac{5}{8} \div \frac{2}{3} + 4^2 - \frac{1}{2}

Work through these problems, applying the principles and steps we've discussed. Check your answers with a calculator or online resource to ensure accuracy.

Conclusion

Evaluating mathematical expressions requires a systematic approach, guided by the order of operations and a solid understanding of fundamental concepts. By breaking down complex expressions into manageable steps, we can arrive at the correct solution with confidence. In this comprehensive guide, we've meticulously evaluated the expression 34+33−12÷223\frac{3}{4}+3^3-\frac{1}{2} \div 2 \frac{2}{3}, expressing the solution as the mixed number 2791627 \frac{9}{16}.

Remember to practice regularly and apply the key concepts we've discussed. With consistent effort, you'll develop your skills in evaluating expressions and excel in mathematics.