1. A) Simplify The Expression: (7/25) * (10/14) * (100/625) B) Simplify The Expression: (4/7) * (11/5) * (5/3) * (7/5) C) Simplify The Expression: (12/11) * ((7/4) * (8/3)) D) Simplify The Expression: -1(5/15) - (32/64) - (1/25)

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This article dives deep into the simplification of complex fraction expressions, providing a step-by-step guide to mastering these essential mathematical operations. We will dissect several examples, each designed to illustrate different techniques and challenges encountered when working with fractions. Understanding these concepts is crucial for building a strong foundation in algebra and beyond.

1. a) Simplifying rac{7}{25} rac{10}{14} rac{100}{625}

Our journey begins with the expression rac{7}{25} rac{10}{14} rac{100}{625}. To effectively simplify this, we'll employ the principle of multiplying fractions, which involves multiplying the numerators (the top numbers) together and the denominators (the bottom numbers) together. Before we jump into the multiplication, a crucial step is to look for opportunities to simplify individual fractions or cross-simplify between fractions. This process significantly reduces the size of the numbers we're dealing with, making the calculation much easier. Let's break it down:

First, observe that rac10}{14} can be simplified by dividing both numerator and denominator by their greatest common divisor (GCD), which is 2. This gives us rac{10 ÷ 2}{14 ÷ 2} = rac{5}{7}. Similarly, the fraction rac{100}{625} can be simplified by dividing both the numerator and the denominator by their GCD, which is 25. This gives us rac{100 ÷ 25}{625 ÷ 25} = rac{4}{25}. Now our expression looks like this rac{7{25} rac{5}{7} rac{4}{25}.

Next, we can look for opportunities to cross-simplify. Notice that the 7 in the numerator of the first fraction and the 7 in the denominator of the second fraction can cancel each other out. Similarly, the 5 in the numerator of the second fraction and the 25 in the denominator of the first fraction can be simplified. Dividing both 5 and 25 by their GCD, which is 5, gives us 1 in the numerator and 5 in the denominator. Our expression now transforms to rac{1}{5} rac{1}{1} rac{4}{25}.

Finally, we multiply the remaining numerators and denominators: (1 1 4) / (5 1 25) = 4 / 125. Therefore, the simplified form of the expression rac{7}{25} rac{10}{14} rac{100}{625} is rac{4}{125}. This meticulous simplification process not only arrives at the correct answer but also demonstrates the power of reducing fractions before multiplying, a technique that saves time and reduces the risk of errors.

1. b) Simplifying rac{4}{7} rac{11}{5} rac{5}{3} rac{7}{5}

Now, let's tackle the expression rac{4}{7} rac{11}{5} rac{5}{3} rac{7}{5}. This expression presents another opportunity to showcase the efficiency of simplifying before multiplying. By carefully observing the fractions, we can identify common factors between numerators and denominators, allowing us to reduce the complexity of the calculation.

Firstly, we notice that the fraction includes the terms rac{11}{5}, rac{5}{3}, and rac{7}{5}. Before performing any multiplication, we should seek out opportunities for cross-cancellation. We can see that the 5 in the numerator of rac{5}{3} and the 5 in the denominator of rac{11}{5} can be simplified. We also observe that the 7 in the denominator of rac{4}{7} and the 7 in the numerator of rac{7}{5} can be simplified. These simplifications significantly reduce the magnitude of the numbers we need to multiply.

After cross-cancellation, the expression is transformed. The 7 in the denominator of the first fraction cancels with the 7 in the numerator of the fourth fraction, leaving 1 in both places. Also, one of the 5s in the numerator cancels with one of the 5s in the denominator, again leaving 1. The expression now looks like this: rac{4}{1} rac{11}{5} rac{1}{3} rac{1}{5}.

However, upon closer inspection, we realize that we missed a crucial simplification opportunity. We have the fraction rac{11}{5} and the original expression had two instances of 5 in the denominator. After the first simplification, we are left with a 5 in the denominator of the third fraction. Let’s correct this and revisit our cancellation. The original expression was rac{4}{7} rac{11}{5} rac{5}{3} rac{7}{5}.

Correcting the simplification, the 7 in the denominator of the first fraction cancels with the 7 in the numerator of the fourth fraction. The 5 in the numerator of the third fraction cancels with the 5 in the denominator of the second fraction. This leaves us with the simplified expression rac{4}{1} rac{11}{1} rac{1}{3} rac{1}{5}.

Now, we multiply the numerators together: 4 11 1 1 = 44. And we multiply the denominators together: 1 1 3 5 = 15. Thus, the simplified fraction is rac{44}{15}. This fraction is already in its simplest form because 44 and 15 have no common factors other than 1. The final answer, therefore, is rac{44}{15}, highlighting the importance of careful and systematic simplification.

1. c) Simplifying rac{12}{11} ( rac{7}{4} rac{8}{3})

Moving on, we encounter the expression rac{12}{11} ( rac{7}{4} rac{8}{3}). This expression introduces parentheses, which dictate the order of operations. According to the order of operations (PEMDAS/BODMAS), we must first simplify the expression within the parentheses before performing any other operations. This example provides an excellent opportunity to reinforce the importance of following the correct order of operations and demonstrates how strategic simplification can streamline complex calculations.

Our initial focus is on simplifying the expression within the parentheses: ( rac{7}{4} rac{8}{3}). Before multiplying these fractions, we look for opportunities to simplify. We can observe that 4 in the denominator of the first fraction and 8 in the numerator of the second fraction share a common factor of 4. Dividing both 4 and 8 by 4 simplifies the fractions, giving us 1 in the denominator of the first fraction and 2 in the numerator of the second fraction. The expression within the parentheses now becomes ( rac{7}{1} rac{2}{3}).

Next, we multiply the simplified fractions within the parentheses: (7 2) / (1 3) = 14 / 3. Now we have simplified the expression within the parentheses to a single fraction, rac{14}{3}. We can now rewrite the original expression as rac{12}{11} rac{14}{3}.

Now we need to multiply these two fractions. Before we do, we once again look for opportunities to simplify. We notice that 12 in the numerator of the first fraction and 3 in the denominator of the second fraction share a common factor of 3. Dividing both 12 and 3 by 3, we get 4 in the numerator of the first fraction and 1 in the denominator of the second fraction. The expression now simplifies to rac{4}{11} rac{14}{1}.

Finally, we multiply the remaining numerators and denominators: (4 14) / (11 1) = 56 / 11. The resulting fraction, rac{56}{11}, is in its simplest form because 56 and 11 have no common factors other than 1. Therefore, the simplified form of the original expression rac{12}{11} ( rac{7}{4} rac{8}{3}) is rac{56}{11}. This example underscores the importance of adhering to the order of operations and the benefits of simplifying fractions at every possible step.

1. d) Simplifying -1 rac{5}{15} - rac{32}{64} - rac{1}{25}

Lastly, we address the expression -1 rac{5}{15} - rac{32}{64} - rac{1}{25}. This expression involves mixed numbers and subtraction of fractions, adding another layer of complexity. To simplify this, we'll need to convert the mixed number to an improper fraction, simplify individual fractions, find a common denominator, and then perform the subtraction. This example will highlight the techniques for dealing with mixed numbers and the importance of finding a common denominator when subtracting fractions.

First, let's convert the mixed number -1 rac{5}{15} to an improper fraction. To do this, we multiply the whole number part (-1) by the denominator (15) and add the numerator (5). This gives us (-1 15) + 5 = -15 + 5 = -10. We then place this result over the original denominator, giving us - rac{10}{15}. Now, we can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 5. This simplifies - rac{10}{15} to - rac{2}{3}.

Next, we simplify the fraction rac32}{64}. Both 32 and 64 are divisible by 32, so we divide both the numerator and the denominator by 32, giving us rac{32 ÷ 32}{64 ÷ 32} = rac{1}{2}. The expression now looks like this - rac{2{3} - rac{1}{2} - rac{1}{25}.

To subtract these fractions, we need to find a common denominator. The least common multiple (LCM) of 3, 2, and 25 is 150. So, we need to convert each fraction to an equivalent fraction with a denominator of 150. For - rac{2}{3}, we multiply both the numerator and the denominator by 50, giving us - rac{2 50}{3 50} = - rac{100}{150}. For rac{1}{2}, we multiply both the numerator and the denominator by 75, giving us rac{1 75}{2 75} = rac{75}{150}. And for rac{1}{25}, we multiply both the numerator and the denominator by 6, giving us rac{1 6}{25 6} = rac{6}{150}.

Now our expression is - rac100}{150} - rac{75}{150} - rac{6}{150}. We can now perform the subtraction -100 - 75 - 6 = -181. So, the result is - rac{181{150}. This fraction is already in its simplest form because 181 is a prime number and does not share any common factors with 150 other than 1. Therefore, the simplified form of the original expression -1 rac{5}{15} - rac{32}{64} - rac{1}{25} is - rac{181}{150}.

In conclusion, simplifying complex fraction expressions requires a combination of skills, including multiplying and dividing fractions, finding common denominators, simplifying fractions, and following the correct order of operations. By breaking down each expression into smaller, manageable steps and systematically applying these techniques, we can effectively simplify even the most daunting fraction problems. Mastering these skills is not only essential for success in mathematics but also provides a foundation for problem-solving in many other areas of life.