Simplify The Expression:${ 2 \sqrt{5}(\sqrt{2} - 2 \sqrt{3}) }$
Simplify the Expression: 2√5(√2 - 2√3)
In this article, we will simplify the given expression, 2√5(√2 - 2√3), using algebraic manipulation and properties of radicals. The expression involves the product of two square roots, and we will use the distributive property to expand and simplify it.
The given expression is a product of two terms: 2√5 and (√2 - 2√3). To simplify this expression, we need to understand the properties of radicals and how to manipulate them. The expression involves the product of two square roots, which can be simplified using the distributive property.
The distributive property states that for any real numbers a, b, and c, the following equation holds:
a(b + c) = ab + ac
We can use this property to expand and simplify the given expression. By distributing 2√5 to both terms inside the parentheses, we get:
2√5(√2 - 2√3) = 2√5(√2) - 2√5(2√3)
Now that we have expanded the expression using the distributive property, we can simplify it further. We can start by simplifying the first term, 2√5(√2):
2√5(√2) = 2√(5 × 2) = 2√10
Next, we can simplify the second term, 2√5(2√3):
2√5(2√3) = 2 × 2 × √(5 × 3) = 4√15
Now that we have simplified both terms, we can combine them to get the final simplified expression:
2√5(√2 - 2√3) = 2√10 - 4√15
In this article, we simplified the given expression, 2√5(√2 - 2√3), using algebraic manipulation and properties of radicals. We used the distributive property to expand and simplify the expression, and then combined the terms to get the final simplified expression. The simplified expression is 2√10 - 4√15.
Radicals are a fundamental concept in algebra, and understanding their properties is essential for simplifying expressions. Some key properties of radicals include:
- Product of Radicals: The product of two radicals can be simplified using the distributive property.
- Quotient of Radicals: The quotient of two radicals can be simplified by dividing the radicands.
- Power of a Radical: A radical raised to a power can be simplified by raising the radicand to that power.
Here are some examples of simplifying expressions using the properties of radicals:
- Example 1: Simplify the expression 3√(4 × 9).
- Solution: Using the property of product of radicals, we can simplify the expression as follows:
3√(4 × 9) = 3√36 = 3 × 6 = 18
- Example 2: Simplify the expression 2√(16 ÷ 4).
- Solution: Using the property of quotient of radicals, we can simplify the expression as follows:
2√(16 ÷ 4) = 2√4 = 2 × 2 = 4
- Example 3: Simplify the expression (√2)³.
- Solution: Using the property of power of a radical, we can simplify the expression as follows:
(√2)³ = (√2) × (√2) × (√2) = 2 × 2 × 2 = 8
Simplifying expressions involving radicals has many real-world applications in fields such as engineering, physics, and computer science. For example:
- Engineering: Simplifying expressions involving radicals can help engineers design and optimize systems, such as electrical circuits and mechanical systems.
- Physics: Simplifying expressions involving radicals can help physicists model and analyze complex phenomena, such as wave propagation and quantum mechanics.
- Computer Science: Simplifying expressions involving radicals can help computer scientists develop efficient algorithms and data structures, such as sorting and searching algorithms.
In this article, we simplified the given expression, 2√5(√2 - 2√3), using algebraic manipulation and properties of radicals. We used the distributive property to expand and simplify the expression, and then combined the terms to get the final simplified expression. The simplified expression is 2√10 - 4√15. We also discussed the properties of radicals and provided examples of simplifying expressions using these properties. Finally, we highlighted the real-world applications of simplifying expressions involving radicals.
Simplify the Expression: 2√5(√2 - 2√3) - Q&A
In our previous article, we simplified the given expression, 2√5(√2 - 2√3), using algebraic manipulation and properties of radicals. We used the distributive property to expand and simplify the expression, and then combined the terms to get the final simplified expression. The simplified expression is 2√10 - 4√15. In this article, we will answer some frequently asked questions related to simplifying expressions involving radicals.
Q: What is the distributive property?
A: The distributive property is a fundamental concept in algebra that states that for any real numbers a, b, and c, the following equation holds:
a(b + c) = ab + ac
This property allows us to expand and simplify expressions involving radicals.
Q: How do I simplify an expression involving radicals?
A: To simplify an expression involving radicals, you can use the distributive property to expand and simplify the expression. Then, combine the terms to get the final simplified expression.
Q: What are some common properties of radicals?
A: Some common properties of radicals include:
- Product of Radicals: The product of two radicals can be simplified using the distributive property.
- Quotient of Radicals: The quotient of two radicals can be simplified by dividing the radicands.
- Power of a Radical: A radical raised to a power can be simplified by raising the radicand to that power.
Q: How do I simplify an expression involving a product of radicals?
A: To simplify an expression involving a product of radicals, you can use the distributive property to expand and simplify the expression. Then, combine the terms to get the final simplified expression.
Q: How do I simplify an expression involving a quotient of radicals?
A: To simplify an expression involving a quotient of radicals, you can divide the radicands to get the final simplified expression.
Q: How do I simplify an expression involving a power of a radical?
A: To simplify an expression involving a power of a radical, you can raise the radicand to that power to get the final simplified expression.
Q: What are some real-world applications of simplifying expressions involving radicals?
A: Simplifying expressions involving radicals has many real-world applications in fields such as engineering, physics, and computer science. For example:
- Engineering: Simplifying expressions involving radicals can help engineers design and optimize systems, such as electrical circuits and mechanical systems.
- Physics: Simplifying expressions involving radicals can help physicists model and analyze complex phenomena, such as wave propagation and quantum mechanics.
- Computer Science: Simplifying expressions involving radicals can help computer scientists develop efficient algorithms and data structures, such as sorting and searching algorithms.
Q: How do I practice simplifying expressions involving radicals?
A: To practice simplifying expressions involving radicals, you can try the following:
- Practice problems: Try simplifying expressions involving radicals using the distributive property and other properties of radicals.
- Real-world applications: Apply the concepts of simplifying expressions involving radicals to real-world problems in fields such as engineering, physics, and computer science.
- Online resources: Use online resources, such as video tutorials and practice problems, to help you practice simplifying expressions involving radicals.
In this article, we answered some frequently asked questions related to simplifying expressions involving radicals. We discussed the distributive property, common properties of radicals, and real-world applications of simplifying expressions involving radicals. We also provided tips on how to practice simplifying expressions involving radicals. By following these tips and practicing regularly, you can become proficient in simplifying expressions involving radicals and apply these concepts to real-world problems.