Simplifying Complex Mathematical Expressions A Step By Step Guide

In the realm of mathematics, simplifying expressions is a fundamental skill that allows us to manipulate equations and formulas into more manageable and understandable forms. It involves applying various mathematical operations and principles to reduce the complexity of an expression while preserving its value. This article delves into the step-by-step simplification of two distinct mathematical expressions, showcasing the application of order of operations, fraction arithmetic, and algebraic manipulation. Mastering these techniques is crucial for students, educators, and anyone working with mathematical concepts, as it lays the foundation for more advanced problem-solving and analytical skills. Through this comprehensive guide, we aim to enhance your understanding and proficiency in simplifying complex expressions, making mathematical tasks more approachable and efficient.

Part a: Simplifying {\frac{4}{5} \times \left[3.4 - \left{\frac{19}{20} + \frac{2}{5} + \left(6 - \frac{2}{5} + \frac{3}{11}\right)\right}\right]}

Step 1: Convert Decimal to Fraction

The first step in simplifying this expression involves converting the decimal number 3.4 into its fractional equivalent. This conversion is essential for maintaining consistency in the expression and facilitating arithmetic operations. The decimal 3.4 can be expressed as a mixed number, which is a combination of a whole number and a proper fraction. To convert 3.4 to a fraction, we recognize that the digit 4 is in the tenths place, so 3.4 can be written as 3 and 4/10. To further simplify, we reduce the fraction 4/10 to its simplest form by dividing both the numerator and the denominator by their greatest common divisor, which is 2. Thus, 4/10 simplifies to 2/5. Therefore, the decimal 3.4 is equivalent to the mixed number 3 2/5. To convert this mixed number to an improper fraction, we multiply the whole number (3) by the denominator (5) and add the numerator (2), placing the result over the original denominator (5). This gives us (3 * 5 + 2) / 5 = 17/5. By converting the decimal to a fraction, we prepare the expression for further simplification using fractional arithmetic rules.

Step 2: Simplify the Innermost Parentheses

Next, we address the innermost parentheses: extbf{(6 - ${\frac{2}{5}}$ + ${\frac{3}{11}}$)}. To simplify this, we first need to find a common denominator for the fractions. The least common multiple (LCM) of 5 and 11 is 55. We then convert each fraction to an equivalent fraction with a denominator of 55.

• ${ \frac{2}{5} = \frac{2 \times 11}{5 \times 11} = \frac{22}{55} }$
• ${ \frac{3}{11} = \frac{3 \times 5}{11 \times 5} = \frac{15}{55} }$

Now we rewrite the expression inside the parentheses with the common denominator:

• ${ 6 - \frac{22}{55} + \frac{15}{55} }$

To combine these terms, we need to express the whole number 6 as a fraction with a denominator of 55. This is done by multiplying 6 by 55, which gives us 330. So, 6 can be written as ${\frac{330}{55}}$. Now we can rewrite the expression as:

• ${ \frac{330}{55} - \frac{22}{55} + \frac{15}{55} }$

Now, we perform the subtraction and addition:

• ${ \frac{330 - 22 + 15}{55} = \frac{323}{55} }$

So, the simplified form of the innermost parentheses is ${\frac{323}{55}}$. This result will be used in the next step to further simplify the overall expression.

Step 3: Simplify the Curly Braces

Having simplified the innermost parentheses, the next step involves dealing with the curly braces: ${\{\frac{19}{20} + \frac{2}{5} + \frac{323}{55}\}}$. To combine these fractions, we need to find the least common multiple (LCM) of the denominators 20, 5, and 55.

The prime factorization of each denominator is:

• 20 = 2^2 * 5
• 5 = 5
• 55 = 5 * 11

To find the LCM, we take the highest power of each prime factor present in the factorizations: 2^2, 5, and 11. Thus, the LCM is 2^2 * 5 * 11 = 4 * 5 * 11 = 220.

Now, we convert each fraction to an equivalent fraction with a denominator of 220:

• ${ \frac{19}{20} = \frac{19 \times 11}{20 \times 11} = \frac{209}{220} }$
• ${ \frac{2}{5} = \frac{2 \times 44}{5 \times 44} = \frac{88}{220} }$
• ${ \frac{323}{55} = \frac{323 \times 4}{55 \times 4} = \frac{1292}{220} }$

Now we rewrite the expression inside the curly braces with the common denominator:

• ${ \frac{209}{220} + \frac{88}{220} + \frac{1292}{220} }$

Next, we add the numerators:

• ${ \frac{209 + 88 + 1292}{220} = \frac{1589}{220} }$

So, the simplified form of the expression inside the curly braces is ${\frac{1589}{220}}$. This result will be used in the next step to further simplify the overall expression by subtracting it from 3.4 (or its fractional equivalent, 17/5).

Step 4: Simplify the Square Brackets

With the expression inside the curly braces simplified to ${\frac{1589}{220}}$, we now focus on the square brackets: ${3.4 - \{\frac{1589}{220}\}}$. Recall that we converted 3.4 to its fractional equivalent, ${\frac{17}{5}}$. So, the expression within the square brackets becomes:

• ${ \frac{17}{5} - \frac{1589}{220} }$

To subtract these fractions, we need to find a common denominator. The least common multiple (LCM) of 5 and 220 is 220. We convert ${\frac{17}{5}}$ to an equivalent fraction with a denominator of 220:

• ${ \frac{17}{5} = \frac{17 \times 44}{5 \times 44} = \frac{748}{220} }$

Now we can rewrite the expression inside the square brackets with the common denominator:

• ${ \frac{748}{220} - \frac{1589}{220} }$

Next, we subtract the numerators:

• ${ \frac{748 - 1589}{220} = \frac{-841}{220} }$

So, the simplified form of the expression inside the square brackets is ${\frac{-841}{220}}$. This result will be used in the final step to complete the simplification of the entire expression.

Step 5: Multiply by ${\frac{4}{5}}$

The final step involves multiplying the result from the square brackets, which is ${\frac{-841}{220}}$, by ${\frac{4}{5}}$. The expression now looks like this:

• ${ \frac{4}{5} \times \frac{-841}{220} }$

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

• ${ \frac{4 \times -841}{5 \times 220} = \frac{-3364}{1100} }$

Now, we simplify the resulting fraction. Both the numerator and the denominator are divisible by 4:

• ${ \frac{-3364 \div 4}{1100 \div 4} = \frac{-841}{275} }$

The fraction ${\frac{-841}{275}}$ is already in its simplest form because 841 and 275 have no common factors other than 1. Therefore, the final simplified form of the given expression is ${\frac{-841}{275}}$. This concludes the simplification of the first part of the problem.

Part b: Simplifying ${2\frac{1}{7} + 9\left[\frac{5}{6} - \frac{1}{9} \left\{\frac{6}{7} \div \left(45 - \frac{1}{7}\right)\right\}\right]}$

Step 1: Convert Mixed Fraction to Improper Fraction

The first step in simplifying this expression involves converting the mixed fraction ${2\frac{1}{7}}$ into an improper fraction. A mixed fraction consists of a whole number and a proper fraction. To convert it to an improper fraction, we multiply the whole number by the denominator of the fractional part and add the numerator. The result is then placed over the original denominator.

In this case, we multiply 2 (the whole number) by 7 (the denominator) and add 1 (the numerator): 2 * 7 + 1 = 14 + 1 = 15. Therefore, the improper fraction equivalent of ${2\frac{1}{7}}$ is ${\frac{15}{7}}$. This conversion is crucial for simplifying the expression as it allows us to perform arithmetic operations more easily with fractions.

Step 2: Simplify the Innermost Parentheses

Moving on to the innermost parentheses, we have ${45 - \frac{1}{7}}$. To subtract a fraction from a whole number, we need to express the whole number as a fraction with the same denominator as the fraction being subtracted. In this case, we need to express 45 as a fraction with a denominator of 7.

To do this, we multiply 45 by 7, which gives us 315. So, 45 can be written as ${\frac{315}{7}}$. Now we can rewrite the expression inside the parentheses as:

• ${ \frac{315}{7} - \frac{1}{7} }$

Subtracting the fractions involves subtracting the numerators while keeping the denominator the same:

• ${ \frac{315 - 1}{7} = \frac{314}{7} }$

Thus, the simplified form of the expression inside the innermost parentheses is ${\frac{314}{7}}$. This result will be used in the next step to further simplify the overall expression by performing the division operation within the curly braces.

Step 3: Perform the Division within Curly Braces

Now, we address the division operation within the curly braces: ${\frac{6}{7} \div \frac{314}{7}}$. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is obtained by swapping its numerator and denominator. Therefore, the reciprocal of ${\frac{314}{7}}$ is ${\frac{7}{314}}$.

So, the division operation becomes a multiplication:

• ${ \frac{6}{7} \div \frac{314}{7} = \frac{6}{7} \times \frac{7}{314} }$

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

• ${ \frac{6 \times 7}{7 \times 314} = \frac{42}{2198} }$

Now, we simplify the resulting fraction by finding the greatest common divisor (GCD) of the numerator and the denominator. Both 42 and 2198 are divisible by 2:

• ${ \frac{42 \div 2}{2198 \div 2} = \frac{21}{1099} }$

The fraction ${\frac{21}{1099}}$ is in its simplest form because 21 and 1099 have no common factors other than 1. Therefore, the simplified form of the expression within the division operation is ${\frac{21}{1099}}$. This result will be used in the next step to continue simplifying the expression within the square brackets.

Step 4: Simplify the Expression within Square Brackets

Having simplified the division within the curly braces, we now focus on the expression within the square brackets: ${\frac{5}{6} - \frac{1}{9} \times \frac{21}{1099}}$. According to the order of operations (PEMDAS/BODMAS), we perform multiplication before subtraction. So, we first multiply ${\frac{1}{9}}$ by ${\frac{21}{1099}}$:

• ${ \frac{1}{9} \times \frac{21}{1099} = \frac{1 \times 21}{9 \times 1099} = \frac{21}{9891} }$

Now, we can simplify the fraction ${\frac{21}{9891}}$. Both the numerator and the denominator are divisible by 3:

• ${ \frac{21 \div 3}{9891 \div 3} = \frac{7}{3297} }$

Now the expression within the square brackets becomes:

• ${ \frac{5}{6} - \frac{7}{3297} }$

To subtract these fractions, we need to find a common denominator. The least common multiple (LCM) of 6 and 3297 is 6594. We convert each fraction to an equivalent fraction with a denominator of 6594:

• ${ \frac{5}{6} = \frac{5 \times 1099}{6 \times 1099} = \frac{5495}{6594} }$
• ${ \frac{7}{3297} = \frac{7 \times 2}{3297 \times 2} = \frac{14}{6594} }$

Now we can rewrite the expression inside the square brackets with the common denominator:

• ${ \frac{5495}{6594} - \frac{14}{6594} }$

Next, we subtract the numerators:

• ${ \frac{5495 - 14}{6594} = \frac{5481}{6594} }$

So, the simplified form of the expression inside the square brackets is ${\frac{5481}{6594}}$. This result will be used in the next step to multiply by 9 and then add to the fraction outside the brackets.

Step 5: Multiply by 9 and Add ${\frac{15}{7}}$

The final steps involve multiplying the result from the square brackets by 9 and then adding the fraction ${\frac{15}{7}}$ to the result. The expression now looks like this:

• ${ \frac{15}{7} + 9 \times \frac{5481}{6594} }$

First, we perform the multiplication:

• ${ 9 \times \frac{5481}{6594} = \frac{9 \times 5481}{6594} = \frac{49329}{6594} }$

Now, we simplify the fraction ${\frac{49329}{6594}}$. Both the numerator and the denominator are divisible by 3:

• ${ \frac{49329 \div 3}{6594 \div 3} = \frac{16443}{2198} }$

Next, we add ${\frac{15}{7}}$ to ${\frac{16443}{2198}}$. To add these fractions, we need to find a common denominator. The least common multiple (LCM) of 7 and 2198 is 15386. We convert each fraction to an equivalent fraction with a denominator of 15386:

• ${ \frac{15}{7} = \frac{15 \times 2198}{7 \times 2198} = \frac{32970}{15386} }$
• ${ \frac{16443}{2198} = \frac{16443 \times 7}{2198 \times 7} = \frac{115101}{15386} }$

Now we can rewrite the expression with the common denominator:

• ${ \frac{32970}{15386} + \frac{115101}{15386} }$

Next, we add the numerators:

• ${ \frac{32970 + 115101}{15386} = \frac{148071}{15386} }$

So, the final simplified form of the given expression is ${\frac{148071}{15386}}$. This concludes the simplification of the second part of the problem.

In summary, the simplification of mathematical expressions involves a systematic approach, adhering to the order of operations and applying relevant arithmetic principles. In the first expression, by converting decimals to fractions, finding common denominators, and carefully performing operations within parentheses, brackets, and braces, we simplified {\frac{4}{5} \times \left[3.4 - \left{\frac{19}{20} + \frac{2}{5} + \left(6 - \frac{2}{5} + \frac{3}{11}\right)\right}\right]} to ${\frac{-841}{275}}$. Similarly, for the second expression, ${2\frac{1}{7} + 9\left[\frac{5}{6} - \frac{1}{9} \left\{\frac{6}{7} \div \left(45 - \frac{1}{7}\right)\right\}\right]}$, we converted mixed fractions to improper fractions, simplified within parentheses and brackets, and followed the order of operations to arrive at the simplified form of ${\frac{148071}{15386}}$. These step-by-step solutions underscore the importance of accuracy, attention to detail, and a solid grasp of mathematical concepts in simplifying complex expressions. By mastering these skills, one can confidently tackle a wide range of mathematical problems and gain a deeper appreciation for the elegance and precision of mathematics.