Solve The Following Mathematical Expressions: a) $7 : (2.5 - \frac{2}{5}) + \frac{11}{30} : \frac{1}{3} + (-1)^{2005}$ b) $3 - \frac{2}{5} \left[ 3 - \frac{2}{5} \left( 1 - \frac{1}{6} \right) \right]$ c) $\left( -\frac{2}{5} \right)^4 : \left( \frac{2}{5} \right)^2 - 8 \cdot \left( 2 - \frac{5}{4} \right)^3$

In the realm of mathematics, the ability to solve complex expressions accurately and efficiently is a cornerstone of problem-solving prowess. This article delves into the intricacies of evaluating mathematical expressions, providing a step-by-step guide to tackle even the most challenging problems. We will dissect three distinct expressions, each presenting a unique set of operations and nuances, to equip you with the skills and confidence to conquer any mathematical hurdle.

Unveiling the Order of Operations: A Foundation for Success

Before we embark on our journey of solving complex expressions, it is imperative 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 hierarchy dictates the sequence in which operations must be performed to arrive at the correct answer.

Parentheses (or brackets) take precedence, meaning any operations enclosed within parentheses should be executed first. This allows us to isolate and simplify specific parts of an expression before addressing the broader calculation.

Exponents come next in the order of operations. Exponents indicate the number of times a base number is multiplied by itself, and their evaluation is crucial before proceeding with other operations.

Multiplication and Division share equal footing in the hierarchy and are performed from left to right. This means that if multiplication and division appear consecutively in an expression, the operation that comes first from left to right should be executed first.

Addition and Subtraction also share equal priority and are performed from left to right, similar to multiplication and division.

By adhering to the order of operations, we ensure consistency and accuracy in our calculations, preventing ambiguity and leading to reliable solutions.

Case Study 1: Tackling a Multi-faceted Expression

Let's begin with our first expression:

a) $7 : (2.5 - \frac{2}{5}) + \frac{11}{30} : \frac{1}{3} + (-1)^{2005}$

This expression presents a blend of arithmetic operations, including division, subtraction, addition, and exponentiation. To solve it effectively, we must meticulously follow the order of operations.

1. Parentheses: Our initial focus is on the expression within the parentheses: $(2.5 - \frac{2}{5})$. To perform this subtraction, we need to express both terms with a common denominator. Converting 2.5 to a fraction, we get $\frac{5}{2}$. Now, we can rewrite the expression as $(\frac{5}{2} - \frac{2}{5})$. The least common denominator for 2 and 5 is 10. Converting the fractions, we have $(\frac{25}{10} - \frac{4}{10})$, which simplifies to $\frac{21}{10}$.

2. Division (first occurrence): Next, we encounter the division operation: $7 : \frac{21}{10}$. Dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, we can rewrite this as $7 \cdot \frac{10}{21}$. Simplifying, we get $\frac{70}{21}$, which further reduces to $\frac{10}{3}$.

3. Division (second occurrence): We proceed to the next division operation: $\frac{11}{30} : \frac{1}{3}$. Again, we multiply by the reciprocal: $\frac{11}{30} \cdot 3$. This simplifies to $\frac{11}{10}$.

4. Exponent: Now, we address the exponent: $(-1)^{2005}$. Any negative number raised to an odd power results in a negative number. Therefore, $(-1)^{2005} = -1$.

5. Addition: Finally, we combine the results of our previous calculations: $\frac{10}{3} + \frac{11}{10} + (-1)$. To add these terms, we need a common denominator, which is 30. Converting the fractions, we get $\frac{100}{30} + \frac{33}{30} - \frac{30}{30}$. Adding the numerators, we arrive at $\frac{103}{30}$.

Therefore, the solution to the expression $7 : (2.5 - \frac{2}{5}) + \frac{11}{30} : \frac{1}{3} + (-1)^{2005}$ is $\frac{103}{30}$.

Case Study 2: Navigating Nested Parentheses

Our second expression introduces nested parentheses, adding another layer of complexity:

b) $3 - \frac{2}{5} \left[ 3 - \frac{2}{5} \left( 1 - \frac{1}{6} \right) \right]$

Nested parentheses require a systematic approach, working from the innermost parentheses outwards. This ensures that we simplify the expression gradually and accurately.

1. Innermost Parentheses: We begin with the innermost parentheses: $(1 - \frac{1}{6})$. To subtract these terms, we need a common denominator, which is 6. Rewriting the expression, we get $(\frac{6}{6} - \frac{1}{6})$, which simplifies to $\frac{5}{6}$.

2. Multiplication (first occurrence): Next, we perform the multiplication within the brackets: $\frac{2}{5} \cdot \frac{5}{6}$. Multiplying the numerators and denominators, we get $\frac{10}{30}$, which simplifies to $\frac{1}{3}$.

3. Subtraction: Now, we address the subtraction within the brackets: $3 - \frac{1}{3}$. To subtract these terms, we need a common denominator, which is 3. Rewriting the expression, we get $\frac{9}{3} - \frac{1}{3}$, which simplifies to $\frac{8}{3}$.

4. Multiplication (second occurrence): We proceed to the next multiplication: $\frac{2}{5} \cdot \frac{8}{3}$. Multiplying the numerators and denominators, we get $\frac{16}{15}$.

5. Subtraction: Finally, we perform the last subtraction: $3 - \frac{16}{15}$. To subtract these terms, we need a common denominator, which is 15. Rewriting the expression, we get $\frac{45}{15} - \frac{16}{15}$, which simplifies to $\frac{29}{15}$.

Therefore, the solution to the expression $3 - \frac{2}{5} \left[ 3 - \frac{2}{5} \left( 1 - \frac{1}{6} \right) \right]$ is $\frac{29}{15}$.

Case Study 3: Mastering Exponents and Fractions

Our third expression combines exponents and fractions, demanding a precise application of the order of operations:

c) $\left( -\frac{2}{5} \right)^4 : \left( \frac{2}{5} \right)^2 - 8 \cdot \left( 2 - \frac{5}{4} \right)^3$

This expression requires careful attention to both exponentiation and fractional arithmetic.

1. Parentheses: We begin by simplifying the expression inside the rightmost parentheses: $(2 - \frac{5}{4})$. To subtract these terms, we need a common denominator, which is 4. Rewriting the expression, we get $(\frac{8}{4} - \frac{5}{4})$, which simplifies to $\frac{3}{4}$.

2. Exponents (first occurrence): Next, we evaluate the first exponent: $\left( -\frac{2}{5} \right)^4$. Raising a negative fraction to an even power results in a positive fraction. Therefore, $\left( -\frac{2}{5} \right)^4 = \frac{16}{625}$.

3. Exponents (second occurrence): We proceed to the second exponent: $\left( \frac{2}{5} \right)^2$. Squaring the fraction, we get $\frac{4}{25}$.

4. Exponents (third occurrence): We evaluate the third exponent: $\left( \frac{3}{4} \right)^3$. Cubing the fraction, we get $\frac{27}{64}$.

5. Division: Now, we perform the division: $\frac{16}{625} : \frac{4}{25}$. Dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, we can rewrite this as $\frac{16}{625} \cdot \frac{25}{4}$. Simplifying, we get $\frac{400}{2500}$, which further reduces to $\frac{4}{25}$.

6. Multiplication: We proceed to the multiplication: $8 \cdot \frac{27}{64}$. Multiplying, we get $\frac{216}{64}$, which simplifies to $\frac{27}{8}$.

7. Subtraction: Finally, we perform the subtraction: $\frac{4}{25} - \frac{27}{8}$. To subtract these fractions, we need a common denominator, which is 200. Converting the fractions, we get $\frac{32}{200} - \frac{675}{200}$. Subtracting the numerators, we arrive at $\frac{-643}{200}$.

Therefore, the solution to the expression $\left( -\frac{2}{5} \right)^4 : \left( \frac{2}{5} \right)^2 - 8 \cdot \left( 2 - \frac{5}{4} \right)^3$ is $\frac{-643}{200}$.

Conclusion: Mastering the Art of Mathematical Expressions

Solving complex mathematical expressions requires a blend of understanding the order of operations, meticulous attention to detail, and proficiency in arithmetic operations. By dissecting these three case studies, we have demonstrated a step-by-step approach to tackling diverse expressions, empowering you to confidently navigate the intricacies of mathematical calculations. Remember, practice is the key to mastery. Embrace the challenge, hone your skills, and unlock the power of mathematical problem-solving.