Simplifying Expressions A Step By Step Guide To Solving 2/3 + 1^3 - 1/3 ÷ 1 1/5

Have you ever encountered a mathematical expression that seemed daunting at first glance? Expressions involving fractions, exponents, and division can sometimes appear complex, but with a systematic approach, they can be simplified with ease. In this comprehensive guide, we will break down the process of simplifying the expression $\frac{2}{3}+1^3-\frac{1}{3} \div 1 \frac{1}{5}$, providing a clear, step-by-step solution. Our goal is not just to arrive at the correct answer but to equip you with the knowledge and skills to tackle similar problems confidently. So, let's embark on this mathematical journey and unravel the intricacies of this expression together.

Understanding the Order of Operations

Before we dive into the specifics of our expression, it's crucial to understand 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 order dictates the sequence in which operations should be performed to ensure a consistent and accurate result. In essence, PEMDAS serves as a roadmap, guiding us through the complexities of mathematical expressions. Ignoring this order can lead to incorrect answers and a misunderstanding of the underlying mathematical principles. Therefore, let's delve deeper into each component of PEMDAS to fully grasp its significance:

• Parentheses: Operations enclosed within parentheses (or brackets) take precedence over all others. This is because parentheses often group terms or operations that need to be treated as a single unit. For example, in the expression 2 x (3 + 4), we first add 3 and 4 within the parentheses, resulting in 7, and then multiply by 2. Parentheses can also be nested, meaning one set of parentheses can be placed inside another. In such cases, we work from the innermost set outwards, ensuring that each operation is performed in the correct sequence.
• Exponents: Exponents represent repeated multiplication and are evaluated after parentheses. An exponent indicates how many times a base number is multiplied by itself. For instance, in the expression $2^3$, the base is 2 and the exponent is 3, meaning 2 is multiplied by itself three times (2 x 2 x 2), resulting in 8. Exponents play a crucial role in various mathematical contexts, from scientific notation to polynomial expressions, and understanding their proper evaluation is essential for mathematical proficiency.
• Multiplication and Division: Multiplication and division are performed from left to right after exponents. These operations are considered to be of equal precedence, meaning neither takes priority over the other. When an expression contains both multiplication and division, we simply proceed from left to right, performing each operation as it appears. For example, in the expression 12 ÷ 3 x 2, we first divide 12 by 3, resulting in 4, and then multiply by 2, giving us 8. This left-to-right approach ensures consistency and avoids ambiguity in mathematical calculations.
• Addition and Subtraction: Addition and subtraction are the final operations to be performed, also from left to right. Similar to multiplication and division, addition and subtraction are of equal precedence. When an expression contains both addition and subtraction, we simply proceed from left to right, performing each operation as it appears. For instance, in the expression 8 + 5 - 3, we first add 8 and 5, resulting in 13, and then subtract 3, giving us 10. By adhering to this left-to-right approach, we ensure that addition and subtraction are performed in the correct order, leading to accurate results.

In summary, the order of operations is a fundamental principle that governs mathematical calculations. By following PEMDAS, we can ensure that expressions are evaluated correctly, leading to accurate results and a deeper understanding of mathematical concepts. Mastering the order of operations is not just about arriving at the right answer; it's about developing a systematic approach to problem-solving that can be applied across various mathematical domains.

Applying PEMDAS to the Expression

Now that we have a solid understanding of the order of operations (PEMDAS), let's apply it to the expression $\frac{2}{3}+1^3-\frac{1}{3} \div 1 \frac{1}{5}$. This expression involves fractions, exponents, division, addition, and subtraction, making it an excellent example to demonstrate the practical application of PEMDAS. By carefully following each step, we can simplify the expression and arrive at the correct answer.

1. Evaluate the Exponent

The first step, according to PEMDAS, is to address any exponents. In our expression, we have $1^3$, which means 1 raised to the power of 3. This is simply 1 multiplied by itself three times: 1 x 1 x 1. The result is 1. So, we can replace $1^3$ with 1 in our expression:

$\frac{2}{3} + 1 - \frac{1}{3} \div 1 \frac{1}{5}$

2. Convert Mixed Number to Improper Fraction

Next, we need to deal with the mixed number $1 \frac{1}{5}$. A mixed number is a combination of a whole number and a fraction, and it's often easier to work with improper fractions in calculations. An improper fraction is a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). To convert a mixed number to an improper fraction, we multiply the whole number by the denominator and add the numerator. This result becomes the new numerator, and the denominator remains the same.

In this case, we have $1 \frac{1}{5}$. Multiplying the whole number (1) by the denominator (5) gives us 5. Adding the numerator (1) to this result gives us 6. So, the improper fraction equivalent of $1 \frac{1}{5}$ is $\frac{6}{5}$. We can now replace $1 \frac{1}{5}$ with $\frac{6}{5}$ in our expression:

$\frac{2}{3} + 1 - \frac{1}{3} \div \frac{6}{5}$

3. Perform Division

According to PEMDAS, division comes before addition and subtraction. We have $\frac{1}{3} \div \frac{6}{5}$ in our expression. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. So, the reciprocal of $\frac{6}{5}$ is $\frac{5}{6}$.

Now, we can rewrite the division as multiplication:

$\frac{1}{3} \div \frac{6}{5} = \frac{1}{3} \times \frac{5}{6}$

To multiply fractions, we multiply the numerators together and the denominators together:

$\frac{1}{3} \times \frac{5}{6} = \frac{1 \times 5}{3 \times 6} = \frac{5}{18}$

We can now replace $\frac{1}{3} \div \frac{6}{5}$ with $\frac{5}{18}$ in our expression:

$\frac{2}{3} + 1 - \frac{5}{18}$

4. Perform Addition and Subtraction

Finally, we are left with addition and subtraction. According to PEMDAS, we perform these operations from left to right. Our expression is now $\frac{2}{3} + 1 - \frac{5}{18}$.

a. Addition

First, let's add $\frac{2}{3}$ and 1. To add a fraction and a whole number, we can rewrite the whole number as a fraction with the same denominator as the other fraction. In this case, we can rewrite 1 as $\frac{3}{3}$:

$\frac{2}{3} + 1 = \frac{2}{3} + \frac{3}{3}$

Now, we can add the fractions by adding the numerators and keeping the same denominator:

$\frac{2}{3} + \frac{3}{3} = \frac{2 + 3}{3} = \frac{5}{3}$

b. Subtraction

Next, we need to subtract $\frac{5}{18}$ from $\frac{5}{3}$. To subtract fractions, they must have a common denominator. The least common multiple (LCM) of 3 and 18 is 18. So, we need to convert $\frac{5}{3}$ to an equivalent fraction with a denominator of 18. To do this, we multiply both the numerator and the denominator of $\frac{5}{3}$ by 6:

$\frac{5}{3} = \frac{5 \times 6}{3 \times 6} = \frac{30}{18}$

Now, we can subtract the fractions:

$\frac{30}{18} - \frac{5}{18} = \frac{30 - 5}{18} = \frac{25}{18}$

5. Express as a Mixed Number

The result we obtained, $\frac{25}{18}$, is an improper fraction. It's often preferable to express the final answer as a mixed number. To convert an improper fraction to a mixed number, we divide the numerator by the denominator. The quotient (the whole number result of the division) becomes the whole number part of the mixed number, the remainder becomes the numerator of the fractional part, and the denominator remains the same.

Dividing 25 by 18, we get a quotient of 1 and a remainder of 7. So, the mixed number equivalent of $\frac{25}{18}$ is $1 \frac{7}{18}$.

Final Answer

Therefore, by systematically applying the order of operations (PEMDAS) and converting between fractions and mixed numbers, we have successfully simplified the expression $\frac{2}{3}+1^3-\frac{1}{3} \div 1 \frac{1}{5}$. The final answer, expressed as a mixed number, is:

$1 \frac{7}{18}$

This step-by-step guide demonstrates the importance of following the order of operations and provides a clear methodology for simplifying complex expressions. By understanding and applying these principles, you can confidently tackle a wide range of mathematical problems.

Common Mistakes to Avoid

When simplifying mathematical expressions, it's easy to make mistakes if you're not careful. A thorough understanding of the order of operations (PEMDAS) is paramount in avoiding these pitfalls. Let's delve into some common errors that students and even seasoned mathematicians sometimes make, and how you can steer clear of them. By recognizing these common mistakes, you can develop a more robust approach to problem-solving and minimize the chances of arriving at an incorrect answer.

1. Ignoring the Order of Operations

Perhaps the most frequent mistake is neglecting the order of operations. As we've emphasized throughout this guide, PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) provides the roadmap for simplifying expressions. Skimming over this order can lead to significant errors. For instance, consider the expression 2 + 3 x 4. If you add 2 and 3 first and then multiply by 4, you'll get 20. However, the correct approach is to multiply 3 and 4 first (resulting in 12) and then add 2, giving you the correct answer of 14. This simple example illustrates the profound impact of adhering to the order of operations. To avoid this mistake, always take a moment to consciously identify the operations in the expression and the order in which they should be performed. It might be helpful to underline or circle each operation as you plan your approach.

2. Incorrectly Handling Division by Fractions

Division involving fractions can be tricky for many. A common mistake is to perform the division directly without inverting and multiplying. Remember, dividing by a fraction is the same as multiplying by its reciprocal. For instance, to solve $\frac{1}{2} \div \frac{3}{4}$, you should not attempt to divide the numerators and denominators directly. Instead, you should invert the second fraction ($\frac{3}{4}$) to get $\frac{4}{3}$ and then multiply: $\frac{1}{2} \times \frac{4}{3}$. This gives you $\frac{4}{6}$, which simplifies to $\frac{2}{3}$. To avoid this mistake, always remember the rule: "Dividing is as easy as pie, flip the second and multiply!" This simple mnemonic can help you recall the correct procedure for dividing fractions.

3. Errors in Converting Mixed Numbers and Improper Fractions

As we saw in our step-by-step solution, mixed numbers and improper fractions play a crucial role in simplifying expressions. Errors in converting between these forms can lead to incorrect results. For example, to convert the mixed number $2 \frac{1}{3}$ to an improper fraction, you need to multiply the whole number (2) by the denominator (3) and add the numerator (1), placing the result over the original denominator. This gives you $\frac{7}{3}$. A common mistake is to simply add the whole number and the numerator, which would incorrectly give you $\frac{3}{3}$. Similarly, when converting an improper fraction back to a mixed number, you need to divide the numerator by the denominator. The quotient becomes the whole number, the remainder becomes the new numerator, and the denominator remains the same. To avoid these mistakes, practice converting between mixed numbers and improper fractions regularly. You can also use visual aids, such as fraction bars or diagrams, to reinforce your understanding of the concepts.

4. Sign Errors

Sign errors are another common pitfall in mathematical calculations. These errors often arise when dealing with negative numbers or when distributing a negative sign across parentheses. For instance, consider the expression 5 - (3 - 2). A common mistake is to simply subtract 3 from 5 and then subtract 2, which would give you 0. However, the correct approach is to distribute the negative sign across the parentheses: 5 - 3 + 2. This gives you the correct answer of 4. Similarly, when multiplying or dividing numbers with different signs, remember that the result is negative. To avoid sign errors, be meticulous in your calculations and pay close attention to the signs of each term. It might be helpful to rewrite expressions to make the signs clearer, especially when dealing with multiple negative signs.

5. Not Simplifying Completely

Finally, failing to simplify an expression completely is a common oversight. This often occurs when working with fractions. For example, if you arrive at the answer $\frac{6}{8}$, it's technically correct, but it's not in its simplest form. Both the numerator and denominator can be divided by 2, resulting in the simplified fraction $\frac{3}{4}$. To avoid this mistake, always check your final answer to see if it can be simplified further. This may involve reducing fractions to their lowest terms or combining like terms in an algebraic expression. A simplified answer is not only mathematically elegant but also easier to work with in subsequent calculations.

By being mindful of these common mistakes and developing strategies to avoid them, you can significantly improve your accuracy and confidence in simplifying mathematical expressions. Remember, practice makes perfect, so work through a variety of problems, and don't be discouraged by errors. Each mistake is an opportunity to learn and refine your skills.

Practice Problems

To solidify your understanding of simplifying expressions and the order of operations, working through practice problems is essential. Here are a few additional problems that will allow you to apply the concepts and techniques we've discussed. By tackling these problems, you'll gain confidence in your ability to simplify complex expressions and identify potential pitfalls. Remember to approach each problem systematically, following the order of operations (PEMDAS) and checking your work carefully. Don't be afraid to make mistakes – they are valuable learning opportunities. The key is to learn from them and refine your problem-solving skills. So, let's put your knowledge to the test and dive into these practice problems.

1. $\frac{1}{2} + 2^2 - \frac{1}{4} \div \frac{1}{2}$
2. $\frac{3}{4} - 1 + \frac{2}{5} \times 1 \frac{1}{4}$
3. $(\frac{1}{3} + \frac{1}{6}) \div \frac{2}{3} - \frac{1}{2}$

By working through these problems and others like them, you'll reinforce your understanding of the order of operations and develop the skills necessary to simplify a wide range of mathematical expressions. Remember to take your time, be patient, and enjoy the process of problem-solving. With practice, you'll become more confident and proficient in your mathematical abilities.