Simplifying Mathematical Expressions A Comprehensive Guide
Hey guys! Today, we're diving deep into the world of simplifying mathematical expressions. This is a crucial skill in mathematics, and mastering it will make your problem-solving journey way smoother. We'll break down several expressions step-by-step, ensuring you grasp the underlying concepts. So, buckle up and let's get started!
a. Simplifying 53.5 - 1.55.52
When it comes to simplifying expressions, especially those involving decimals and multiple operations, it’s essential to follow the order of operations, often remembered by the acronym PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). In our first expression, 53.5 - 1.55.52, we need to identify the operations involved and their correct sequence. The expression includes subtraction and multiplication. According to the order of operations, multiplication should be performed before subtraction. So, our first task is to multiply 1.55 by 52. This multiplication will give us a product that we will then subtract from 53.5. Let’s break it down step-by-step to make sure we don’t miss anything.
First, we calculate the product of 1.55 and 52. This can be done manually or with a calculator. When you multiply these two numbers, you get 80.6. Now that we have this product, we can rewrite the original expression as 53.5 - 80.6. The next step is to perform the subtraction. Subtracting 80.6 from 53.5 might seem a bit tricky because the number we are subtracting is larger than the number we are subtracting from. This means our result will be a negative number. To perform this subtraction, you essentially find the difference between 80.6 and 53.5 and then apply the negative sign. The difference between 80.6 and 53.5 is 27.1. Therefore, the result of 53.5 - 80.6 is -27.1. This is our simplified answer for the first expression. Understanding the order of operations is crucial in simplifying mathematical expressions, and in this case, multiplying before subtracting was key to arriving at the correct answer. So, the simplified form of 53.5 - 1.55.52 is -27.1. Remember, paying close attention to the signs and the sequence of operations is vital to ensure accuracy.
b. Simplifying 65.(62 : 63)
Moving on to our second expression, 65.(62 : 63), we again need to apply the order of operations to simplify it correctly. Here, we have a combination of multiplication and division. The expression is 65 multiplied by the result of 62 divided by 63. According to PEMDAS/BODMAS, multiplication and division have the same priority, so we perform them from left to right. In this case, we first handle the division inside the parentheses, which is 62 divided by 63, and then multiply the result by 65. This approach ensures we follow the correct mathematical procedure and arrive at the accurate simplified form.
First, let's tackle the division part: 62 divided by 63. This fraction, 62/63, is already in its simplest form since 62 and 63 do not share any common factors other than 1. The division results in a decimal value, which is approximately 0.9841 when rounded to four decimal places. Now that we have the result of the division, we can multiply it by 65. So, we multiply 65 by 0.9841. This multiplication will give us the final simplified value of the expression. Multiplying 65 by 0.9841 gives us approximately 63.9665. Depending on the level of precision required, we might round this number. If we round it to two decimal places, we get 63.97. Therefore, the simplified form of 65.(62 : 63) is approximately 63.97. This process highlights the importance of performing operations in the correct order and the usefulness of calculators or manual calculations to handle decimal values accurately.
c. Simplifying 52. (123) (125):25
Now, let's tackle the expression 52. (123) (125):25. This one looks a bit more complex, involving multiplication and division with multiple numbers. Remember, the key to simplifying such expressions is to follow the order of operations meticulously. In this case, we have a series of multiplications and a division. According to the order of operations, we perform these operations from left to right. This means we will first multiply 52 by 123, then multiply the result by 125, and finally divide the product by 25. By breaking down the expression into these steps, we can ensure a clear and accurate simplification process.
First, we multiply 52 by 123. This gives us a product of 6396. Now, we take this result and multiply it by 125. So, we multiply 6396 by 125, which results in 799500. Finally, we divide this large number by 25. Dividing 799500 by 25 gives us 31980. Therefore, the simplified form of the expression 52. (123) (125):25 is 31980. This example illustrates how breaking down a complex expression into smaller, manageable steps can make the simplification process much easier and less prone to errors. Each multiplication and the final division were performed sequentially to arrive at the final simplified value.
d. Simplifying 10.000 Discussion Category
Lastly, we have the expression 10.000 Discussion Category. This appears to be a slightly different kind of expression, as it includes the term
