7√3-5√2\√48+√18 Simplify
Introduction
Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will focus on simplifying the expression 7√3 - 5√2 √48 + √18. We will break down the expression into smaller parts, simplify each part, and then combine them to get the final result.
Understanding Radical Expressions
Before we dive into simplifying the expression, let's quickly review what radical expressions are. A radical expression is an expression that contains a square root or a higher-order root of a number. The symbol √ is used to represent the square root of a number. For example, √4 = 2, and √9 = 3.
Breaking Down the Expression
Let's break down the expression 7√3 - 5√2 √48 + √18 into smaller parts:
- 7√3
- -5√2
- √48
- √18
Simplifying Each Part
7√3
The expression 7√3 is already simplified, as 7 is a constant and √3 is a radical expression. We can leave it as is.
-5√2
The expression -5√2 is also already simplified, as -5 is a constant and √2 is a radical expression. We can leave it as is.
√48
To simplify √48, we need to find the largest perfect square that divides 48. In this case, 48 = 16 × 3, and √16 = 4. Therefore, √48 = √(16 × 3) = 4√3.
√18
To simplify √18, we need to find the largest perfect square that divides 18. In this case, 18 = 9 × 2, and √9 = 3. Therefore, √18 = √(9 × 2) = 3√2.
Combining the Simplified Parts
Now that we have simplified each part, let's combine them to get the final result:
7√3 - 5√2 4√3 + 3√2
To combine the terms, we need to find a common denominator. In this case, the common denominator is √6. Therefore, we can rewrite the expression as:
7√3 - 5√2 4√3 + 3√2 = 7√3 - 5√6 + 12√6 + 3√2
Now, we can combine the like terms:
7√3 + 7√6 + 3√2
Final Result
The final result is:
7√3 + 7√6 + 3√2
This is the simplified expression 7√3 - 5√2 √48 + √18.
Conclusion
Simplifying radical expressions requires a deep understanding of the underlying mathematics. By breaking down the expression into smaller parts, simplifying each part, and then combining them, we can arrive at the final result. In this article, we simplified the expression 7√3 - 5√2 √48 + √18 and arrived at the final result 7√3 + 7√6 + 3√2. We hope this article has provided valuable insights into simplifying radical expressions and has helped you to improve your math skills.
Additional Resources
If you are struggling with simplifying radical expressions, here are some additional resources that may help:
- Khan Academy: Radical Expressions
- Mathway: Simplifying Radical Expressions
- Wolfram Alpha: Simplifying Radical Expressions
Frequently Asked Questions
Q: What is a radical expression? A: A radical expression is an expression that contains a square root or a higher-order root of a number.
Q: How do I simplify a radical expression? A: To simplify a radical expression, you need to break it down into smaller parts, simplify each part, and then combine them.
Q: What is the common denominator for combining like terms? A: The common denominator is the smallest number that all the terms can be divided by.
Q: What is a radical expression?
A: A radical expression is an expression that contains a square root or a higher-order root of a number. The symbol √ is used to represent the square root of a number. For example, √4 = 2, and √9 = 3.
Q: How do I simplify a radical expression?
A: To simplify a radical expression, you need to break it down into smaller parts, simplify each part, and then combine them. Here are the steps to simplify a radical expression:
- Break down the expression: Break down the expression into smaller parts, such as individual terms or factors.
- Simplify each part: Simplify each part of the expression, using the rules of arithmetic and the properties of radicals.
- Combine the simplified parts: Combine the simplified parts to get the final result.
Q: What is the difference between a radical and a rational number?
A: A radical is an expression that contains a square root or a higher-order root of a number, while a rational number is a number that can be expressed as the ratio of two integers. For example, √4 is a radical, while 2/3 is a rational number.
Q: How do I simplify a radical expression with multiple terms?
A: To simplify a radical expression with multiple terms, you need to combine the terms using the rules of arithmetic and the properties of radicals. Here are the steps to simplify a radical expression with multiple terms:
- Combine like terms: Combine the terms that have the same radical expression.
- Simplify each term: Simplify each term using the rules of arithmetic and the properties of radicals.
- Combine the simplified terms: Combine the simplified terms to get the final result.
Q: What is the common denominator for combining like terms?
A: The common denominator is the smallest number that all the terms can be divided by. For example, if you have two terms with radical expressions √4 and √9, the common denominator is √36, since both terms can be divided by √36.
Q: How do I find the largest perfect square that divides a number?
A: To find the largest perfect square that divides a number, you need to factor the number into its prime factors and then find the largest perfect square that can be formed from those factors. For example, if you have the number 48, you can factor it into 16 × 3, and then find the largest perfect square that can be formed from those factors, which is 16.
Q: What are some common mistakes to avoid when simplifying radical expressions?
A: Here are some common mistakes to avoid when simplifying radical expressions:
- Not simplifying each term: Make sure to simplify each term using the rules of arithmetic and the properties of radicals.
- Not combining like terms: Make sure to combine the terms that have the same radical expression.
- Not using the correct common denominator: Make sure to use the correct common denominator when combining like terms.
- Not checking the final result: Make sure to check the final result to ensure that it is correct.
Q: How can I practice simplifying radical expressions?
A: Here are some ways to practice simplifying radical expressions:
- Work on practice problems: Work on practice problems to get a feel for simplifying radical expressions.
- Use online resources: Use online resources, such as Khan Academy or Mathway, to practice simplifying radical expressions.
- Ask a teacher or tutor: Ask a teacher or tutor for help with simplifying radical expressions.
- Join a study group: Join a study group to practice simplifying radical expressions with others.
Q: What are some real-world applications of simplifying radical expressions?
A: Here are some real-world applications of simplifying radical expressions:
- Engineering: Simplifying radical expressions is used in engineering to solve problems involving geometry and trigonometry.
- Physics: Simplifying radical expressions is used in physics to solve problems involving motion and energy.
- Computer Science: Simplifying radical expressions is used in computer science to solve problems involving algorithms and data structures.
- Finance: Simplifying radical expressions is used in finance to solve problems involving interest rates and investments.
Conclusion
Simplifying radical expressions is an important skill that is used in a variety of fields. By following the steps outlined in this article, you can simplify radical expressions and solve problems involving geometry and trigonometry. Remember to practice simplifying radical expressions regularly to build your skills and confidence.