Simplifying Radicals And Fractions Evaluating √5/6 + √5/10 + 3/8
In the realm of mathematics, we often encounter expressions that combine radicals and fractions, requiring us to apply various arithmetic principles to simplify and understand them. This article delves into the expression √5/6 + √5/10 + 3/8, providing a comprehensive exploration of its components and the methods used to arrive at a simplified form. We will break down each term, discuss the relevant mathematical concepts, and guide you through the step-by-step process of solving this problem. This exploration aims to enhance your understanding of handling such expressions, which are commonly encountered in algebra and calculus.
To effectively tackle √5/6 + √5/10 + 3/8, it’s essential to understand each term individually. We have two terms involving the square root of 5 divided by fractions and a simple fraction.
- √5/6: This term represents the square root of 5 divided by 6. The square root of 5 (approximately 2.236) is an irrational number, meaning it cannot be expressed as a simple fraction. When we divide this by 6, we get another irrational number. Understanding the nature of irrational numbers is crucial when dealing with radicals.
- √5/10: This term is similar to the first but with a different denominator. The square root of 5 is again divided, this time by 10. This fraction can be simplified by recognizing that 10 is twice 5, and this simplification will be a key step in the overall solution.
- 3/8: This is a simple fraction representing three-eighths. It is a rational number and can be expressed as a decimal (0.375). This term is straightforward and will primarily be involved in the final steps of addition.
Now, let's dive into the process of solving √5/6 + √5/10 + 3/8. We will break it down into manageable steps.
Step 1: Simplify the Radical Terms
The first step involves simplifying the terms containing the square root of 5. Both √5/6 and √5/10 have √5 in the numerator, so we can focus on the denominators. It’s beneficial to find a common denominator for the fractions 1/6 and 1/10 to combine these terms effectively. The least common multiple (LCM) of 6 and 10 is 30, so we will rewrite the fractions with this common denominator. This process allows us to combine like terms, making the expression easier to handle. By finding a common denominator, we are essentially preparing the fractions for addition, which is a fundamental arithmetic operation.
Step 2: Finding a Common Denominator
To add √5/6 and √5/10, we need to express both fractions with the same denominator. As mentioned earlier, the least common multiple of 6 and 10 is 30. We convert each fraction as follows:
- √5/6 becomes (5√5)/30 (multiply both numerator and denominator by 5).
- √5/10 becomes (3√5)/30 (multiply both numerator and denominator by 3).
Now we have the fractions with a common denominator, which allows us to add them together seamlessly. This step is crucial because it aligns with the basic principles of fraction addition, where fractions must have the same denominator before they can be combined. The common denominator acts as a standard unit, enabling us to add the numerators directly.
Step 3: Adding the Radical Terms
With the common denominator in place, we can add the radical terms:
(5√5)/30 + (3√5)/30 = (5√5 + 3√5)/30
Here, we add the numerators while keeping the denominator constant. This is a straightforward application of the rules for adding fractions. Now, we combine the terms in the numerator:
5√5 + 3√5 = 8√5
So, the sum becomes (8√5)/30. This simplification demonstrates the distributive property in reverse, where we factor out the common √5 term. The result is a more compact form of the expression, making it easier to work with in subsequent steps.
Step 4: Simplify the Resulting Fraction
Now we simplify the fraction (8√5)/30. Both 8 and 30 are divisible by 2, so we can reduce the fraction:
(8√5)/30 = (4√5)/15
This simplification involves dividing both the numerator and the denominator by their greatest common divisor, which is 2 in this case. Simplifying fractions is a fundamental step in mathematical problem-solving, ensuring that the final answer is expressed in its simplest form. By reducing the fraction, we make it easier to understand and compare with other fractions.
Step 5: Adding the Remaining Fraction
We now need to add the remaining fraction, 3/8, to our simplified radical term (4√5)/15. To do this, we need to find a common denominator for 15 and 8. The least common multiple (LCM) of 15 and 8 is 120. Thus, we convert both fractions to have this denominator.
Step 6: Finding a Common Denominator for 15 and 8
As mentioned, the LCM of 15 and 8 is 120. We convert each fraction as follows:
- (4√5)/15 becomes (32√5)/120 (multiply both numerator and denominator by 8).
- 3/8 becomes 45/120 (multiply both numerator and denominator by 15).
Now, both fractions have the same denominator, allowing us to add them together. This step is crucial for combining the radical term and the rational fraction. The common denominator provides a standardized unit, enabling us to perform the addition accurately.
Step 7: Adding the Fractions
With the common denominator, we can add the fractions:
(32√5)/120 + 45/120 = (32√5 + 45)/120
Here, we add the numerators while keeping the denominator constant. This is a direct application of the rules for adding fractions. The result is a single fraction that combines the radical term and the rational number. This step brings us closer to the final simplified form of the expression.
Step 8: Final Result
The final simplified expression is:
(32√5 + 45)/120
This is the most simplified form of the given expression. We have combined the radical terms, simplified the fractions, and arrived at a final answer that represents the sum of √5/6 + √5/10 + 3/8. This result showcases the combination of a radical term and a rational number, expressed as a single fraction. The numerator contains both a radical and a whole number, while the denominator is a whole number. This form is typical when dealing with expressions that mix radicals and fractions, and it represents the most concise way to express the sum.
While the simplified form (32√5 + 45)/120 is mathematically correct and often preferred for its precision, there are alternative ways to represent this value, such as decimal approximations. These representations can provide a more intuitive sense of the magnitude of the number, especially in practical applications where an exact value may not be necessary.
Decimal Approximation
To obtain a decimal approximation, we can substitute the approximate value of √5 (approximately 2.236) into the expression:
(32 * 2.236 + 45)/120
Calculating this, we get:
(71.552 + 45)/120 = 116.552/120
Dividing 116.552 by 120, we get approximately 0.971.
So, the decimal approximation of (32√5 + 45)/120 is about 0.971. This representation can be useful for quick estimations and comparisons with other numerical values. Decimal approximations are commonly used in fields such as engineering and physics, where precise values may not always be necessary, and a general sense of magnitude is sufficient.
Significance of Approximation
It’s important to note that while the decimal approximation provides a practical value, it is not as precise as the simplified fraction form. The decimal representation is a rounded value, and some information is lost in the rounding process. Therefore, for mathematical proofs and exact calculations, the simplified fraction form is preferred. However, for real-world applications where a rough estimate is sufficient, the decimal approximation is a valuable tool.
In this article, we have thoroughly explored the expression √5/6 + √5/10 + 3/8, breaking it down into individual components and systematically solving it step by step. We began by understanding the nature of each term, including the radical terms and the simple fraction. We then proceeded to simplify the radical terms by finding a common denominator and combining them. The resulting fraction was further simplified before adding the remaining fraction. The final simplified expression, (32√5 + 45)/120, represents the sum in its most concise form. Additionally, we explored the decimal approximation of the expression, providing a practical understanding of its numerical value.
This exploration highlights the importance of understanding fundamental arithmetic operations, such as finding common denominators, simplifying fractions, and combining like terms. These skills are crucial for tackling more complex mathematical problems. By mastering these concepts, you can confidently approach expressions involving radicals and fractions, simplifying them to their most manageable forms. Furthermore, the ability to convert between exact forms and decimal approximations enhances your problem-solving toolkit, allowing you to choose the most appropriate representation for a given context.
Whether you are a student learning algebra or someone interested in refreshing your mathematical skills, this article provides a comprehensive guide to solving expressions of this type. The step-by-step approach, detailed explanations, and alternative representations aim to solidify your understanding and boost your confidence in handling mathematical challenges.
