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

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In this comprehensive guide, we will simplify the expression $\frac{2}{3}+5^2-\frac{2}{3} \div 3 \frac{1}{3}$ step by step. Mathematics often involves dealing with complex expressions, and the key to solving them lies in understanding the order of operations and applying the correct techniques. This article aims to provide a clear, detailed solution, making it easy for anyone to follow along and learn. We'll break down each step, explaining the logic and mathematical principles involved, ensuring that you not only get the correct answer but also understand the process behind it. The final answer will be presented as a mixed number or a fraction in its lowest terms, adhering to standard mathematical practices.

Understanding the Order of Operations

Before diving into the specifics of our expression, it's crucial to understand 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 mathematical operations should be performed to ensure a correct solution. PEMDAS helps to avoid ambiguity and ensures that everyone arrives at the same answer when simplifying expressions. Neglecting the order of operations can lead to drastically different results, highlighting the importance of adhering to this fundamental principle.

1. Parentheses: Operations inside parentheses are always performed first. This includes any kind of grouping symbol like brackets and braces.
2. Exponents: Next, we handle exponents, which indicate repeated multiplication.
3. Multiplication and Division: These operations are performed from left to right.
4. Addition and Subtraction: Finally, addition and subtraction are done from left to right.

By following PEMDAS, we can methodically simplify complex expressions, breaking them down into manageable steps. In our given expression, we will first address the exponent, then the division, and finally the addition and subtraction, ensuring we adhere to this fundamental order.

Step 1: Evaluate the Exponent

The first step in simplifying our expression $\frac{2}{3}+5^2-\frac{2}{3} \div 3 \frac{1}{3}$ is to evaluate the exponent. The term $5^2$ represents 5 raised to the power of 2, which means 5 multiplied by itself. This is a straightforward calculation, and understanding exponents is crucial for more complex mathematical problems. Exponents are a shorthand way of expressing repeated multiplication, and mastering them is essential for algebra, calculus, and other advanced mathematical topics. The ability to quickly and accurately evaluate exponents is a valuable skill in mathematics.

$5^2 = 5 \times 5 = 25$

Now, we can substitute this value back into our original expression:

$\frac{2}{3} + 25 - \frac{2}{3} \div 3 \frac{1}{3}$

This substitution simplifies the expression and allows us to proceed with the next operation according to PEMDAS. By addressing the exponent first, we reduce the complexity of the expression, making it easier to manage in subsequent steps.

Step 2: Convert the Mixed Number to an Improper Fraction

Before we can perform the division, we need to convert the mixed number $3 \frac{1}{3}$ into an improper fraction. A mixed number consists of a whole number and a fraction, while an improper fraction has a numerator that is greater than or equal to its denominator. Converting mixed numbers to improper fractions is essential for performing arithmetic operations like multiplication and division. This conversion ensures that we are working with a single fractional value, simplifying the calculations. Improper fractions are often easier to manipulate in mathematical expressions, making conversions a common practice in arithmetic and algebra.

To convert $3 \frac{1}{3}$ to an improper fraction, we multiply the whole number (3) by the denominator (3) and then add the numerator (1). This result becomes the new numerator, and we keep the original denominator.

$3 \frac{1}{3} = \frac{(3 \times 3) + 1}{3} = \frac{9 + 1}{3} = \frac{10}{3}$

Now, we can substitute this improper fraction back into our expression:

$\frac{2}{3} + 25 - \frac{2}{3} \div \frac{10}{3}$

This substitution prepares us for the division operation, which is the next step in simplifying the expression according to PEMDAS. Converting to improper fractions ensures accuracy and simplifies the division process.

Step 3: Perform the Division

Next, we perform the division operation: $\frac{2}{3} \div \frac{10}{3}$. Dividing fractions involves multiplying by the reciprocal of the divisor. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. This method transforms division into multiplication, which is often easier to handle. Understanding reciprocals and their role in division is a fundamental concept in fraction arithmetic. This step is crucial in simplifying expressions involving fractions, as it allows us to combine terms and move towards a solution.

To divide $\frac{2}{3}$ by $\frac{10}{3}$, we multiply $\frac{2}{3}$ by the reciprocal of $\frac{10}{3}$, which is $\frac{3}{10}$:

$\frac{2}{3} \div \frac{10}{3} = \frac{2}{3} \times \frac{3}{10}$

Now, we multiply the numerators and the denominators:

$\frac{2 \times 3}{3 \times 10} = \frac{6}{30}$

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 6:

$\frac{6}{30} = \frac{6 \div 6}{30 \div 6} = \frac{1}{5}$

Now, we substitute this simplified value back into our expression:

$\frac{2}{3} + 25 - \frac{1}{5}$

This substitution reduces the expression to addition and subtraction operations, which are the final steps in simplifying the original expression.

Step 4: Perform the Addition and Subtraction

Now we are left with addition and subtraction: $\frac{2}{3} + 25 - \frac{1}{5}$. To perform these operations, we need to find a common denominator for the fractions. The common denominator allows us to combine the fractions and simplify the expression further. Finding the least common multiple (LCM) of the denominators is the most efficient way to determine the common denominator. This step is crucial for accurately adding and subtracting fractions and ensuring the final result is in its simplest form.

The denominators are 3 and 5. The least common multiple (LCM) of 3 and 5 is 15. So, we will convert each fraction to have a denominator of 15.

For $\frac{2}{3}$, we multiply both the numerator and denominator by 5:

$\frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}$

For $\frac{1}{5}$, we multiply both the numerator and denominator by 3:

$\frac{1}{5} = \frac{1 \times 3}{5 \times 3} = \frac{3}{15}$

We also need to express 25 as a fraction with a denominator of 15:

$25 = \frac{25}{1} = \frac{25 \times 15}{1 \times 15} = \frac{375}{15}$

Now, we can rewrite our expression with the common denominator:

$\frac{10}{15} + \frac{375}{15} - \frac{3}{15}$

Now, we perform the addition and subtraction:

$\frac{10 + 375 - 3}{15} = \frac{382}{15}$

Step 5: Express the Result as a Mixed Number

The final step is to express the improper fraction $\frac{382}{15}$ as a mixed number. Mixed numbers are often preferred for representing quantities, as they provide a clearer sense of the whole number and fractional parts. Converting improper fractions to mixed numbers involves dividing the numerator by the denominator and expressing the remainder as a fraction. This conversion is a standard practice in mathematics, allowing for easier interpretation and application of fractional values.

To convert $\frac{382}{15}$ to a mixed number, we divide 382 by 15:

$382 \div 15 = 25$ with a remainder of 7.

So, the mixed number is 25 and $\frac{7}{15}$:

$\frac{382}{15} = 25 \frac{7}{15}$

Therefore, the simplified expression is $25 \frac{7}{15}$.

Final Answer

In conclusion, after carefully following the order of operations and simplifying each step, we have determined that:

$\frac{2}{3}+5^2-\frac{2}{3} \div 3 \frac{1}{3} = 25 \frac{7}{15}$

This final answer is presented as a mixed number in its simplest form. By understanding and applying the principles of PEMDAS and fraction arithmetic, we have successfully simplified a complex expression and arrived at the correct solution. This process highlights the importance of methodical problem-solving in mathematics and provides a clear example of how to approach similar problems in the future. Whether you're a student learning these concepts or someone looking to refresh your math skills, this step-by-step guide offers a valuable resource for understanding and mastering the simplification of mathematical expressions.