Simplify The Expression: 5√12 - 19√3

May 4, 2025 by ADMIN 37 views

Simplify the Expression: 5√12 - 19√3

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Introduction

In mathematics, simplifying expressions is an essential skill that helps us solve problems more efficiently. When dealing with square roots, we often encounter expressions that can be simplified using various techniques. In this article, we will focus on simplifying the expression 5√12 - 19√3.

Understanding the Expression

The given expression is 5√12 - 19√3. To simplify this expression, we need to understand the properties of square roots. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, √16 = 4 because 4 × 4 = 16.

Breaking Down the Expression

Let's break down the expression 5√12 - 19√3 into smaller parts. We can start by simplifying the square roots individually.

Simplifying √12

To simplify √12, we need to find the largest perfect square that divides 12. In this case, 12 can be expressed as 4 × 3. Since √4 = 2, we can rewrite √12 as √(4 × 3) = 2√3.

Simplifying √3

The square root of 3 is already in its simplest form, so we don't need to simplify it further.

Simplifying the Expression

Now that we have simplified the individual square roots, we can substitute them back into the original expression.

5√12 - 19√3 = 5(2√3) - 19√3

Using the distributive property, we can expand the expression:

10√3 - 19√3

Combining Like Terms

Now that we have the expression in the form 10√3 - 19√3, we can combine like terms. Since both terms have the same variable (√3), we can add or subtract their coefficients.

10√3 - 19√3 = (10 - 19)√3

Final Simplification

The final simplification of the expression is:

-9√3

Therefore, the simplified expression 5√12 - 19√3 is -9√3.

Conclusion

Tips and Tricks

When simplifying expressions, always look for perfect squares that divide the number under the square root.
Use the distributive property to expand expressions and combine like terms.
Simplify individual square roots before substituting them back into the original expression.

Real-World Applications

Simplifying expressions has numerous real-world applications in fields such as physics, engineering, and computer science. For example, in physics, simplifying expressions helps us solve problems related to motion, energy, and momentum. In engineering, simplifying expressions helps us design and optimize systems, such as bridges and buildings. In computer science, simplifying expressions helps us develop and data structures that are efficient and scalable.

Common Mistakes to Avoid

When simplifying expressions, there are several common mistakes to avoid:

Not simplifying individual square roots before substituting them back into the original expression.
Not using the distributive property to expand expressions and combine like terms.
Not checking for perfect squares that divide the number under the square root.

Final Thoughts

Simplifying expressions is an essential skill in mathematics that helps us solve problems more efficiently. By understanding the properties of square roots and breaking down complex expressions into smaller parts, we can simplify them using various techniques. In this article, we simplified the expression 5√12 - 19√3 using the distributive property and combining like terms. We hope that this article has provided you with a better understanding of how to simplify expressions and has helped you develop your problem-solving skills.

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Introduction

In our previous article, we simplified the expression 5√12 - 19√3 using various techniques. In this article, we will answer some frequently asked questions related to simplifying expressions, with a focus on square roots.

Q&A

Q: What is the difference between a perfect square and a non-perfect square?

A: A perfect square is a number that can be expressed as the product of an integer with itself. For example, 16 is a perfect square because it can be expressed as 4 × 4. A non-perfect square is a number that cannot be expressed as the product of an integer with itself.

Q: How do I simplify a square root of a number that is not a perfect square?

A: To simplify a square root of a number that is not a perfect square, you need to find the largest perfect square that divides the number. For example, to simplify √12, you can express it as √(4 × 3) = 2√3.

Q: What is the distributive property, and how do I use it to simplify expressions?

A: The distributive property is a mathematical property that allows you to expand an expression by multiplying each term inside the parentheses with the term outside the parentheses. For example, to simplify 5(2√3) - 19√3, you can use the distributive property to expand the expression as 10√3 - 19√3.

Q: How do I combine like terms in an expression?

A: To combine like terms in an expression, you need to add or subtract the coefficients of the terms with the same variable. For example, to simplify 10√3 - 19√3, you can combine the like terms to get (10 - 19)√3 = -9√3.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

Not simplifying individual square roots before substituting them back into the original expression.
Not using the distributive property to expand expressions and combine like terms.
Not checking for perfect squares that divide the number under the square root.

Q: How do I apply simplifying expressions in real-world scenarios?

A: Simplifying expressions has numerous real-world applications in fields such as physics, engineering, and computer science. For example, in physics, simplifying expressions helps us solve problems related to motion, energy, and momentum. In engineering, simplifying expressions helps us design and optimize systems, such as bridges and buildings. In computer science, simplifying expressions helps us develop and data structures that are efficient and scalable.

Tips and Tricks

Always look for perfect squares that divide the number under the square root.
Use the distributive property to expand expressions and combine like terms.
Simplify individual square roots before substituting them back into the original expression.
Check for perfect squares that divide the number under the square root.

Real-World Applications

Simplifying expressions has numerous real-world applications in fields such as physics, engineering, and computer science. For example, in physics, simplifying expressions helps us solve problems related to motion energy, and momentum. In engineering, simplifying expressions helps us design and optimize systems, such as bridges and buildings. In computer science, simplifying expressions helps us develop and data structures that are efficient and scalable.

Conclusion

Simplifying expressions is an essential skill in mathematics that helps us solve problems more efficiently. By understanding the properties of square roots and breaking down complex expressions into smaller parts, we can simplify them using various techniques. In this article, we answered some frequently asked questions related to simplifying expressions, with a focus on square roots. We hope that this article has provided you with a better understanding of how to simplify expressions and has helped you develop your problem-solving skills.