Simplify The Expression: ${ 14 \sqrt{20} - 6 \sqrt{5} = }$

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Introduction

Simplifying expressions involving square roots is a crucial skill in mathematics, particularly in algebra and geometry. In this article, we will focus on simplifying the given expression: $14 \sqrt{20} - 6 \sqrt{5}$. We will break down the steps involved in simplifying this expression and provide a clear understanding of the concepts used.

Understanding Square Roots

Before we dive into simplifying the expression, let's quickly review what square roots are. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16. We can represent square roots using the symbol $\sqrt{}$. For instance, $\sqrt{16}$ represents the square root of 16.

Simplifying the Expression

To simplify the expression $14 \sqrt{20} - 6 \sqrt{5}$, we need to start by simplifying the square roots individually. Let's begin with $\sqrt{20}$.

Simplifying $\sqrt{20}$

We can simplify $\sqrt{20}$ by breaking it down into its prime factors. The prime factorization of 20 is $2^2 \times 5$. Therefore, we can rewrite $\sqrt{20}$ as $\sqrt{2^2 \times 5}$.

Using the Property of Square Roots

We know that the square root of a product is equal to the product of the square roots. Using this property, we can rewrite $\sqrt{2^2 \times 5}$ as $\sqrt{2^2} \times \sqrt{5}$.

Simplifying $\sqrt{2^2}$

The square root of $2^2$ is simply 2, because 2 multiplied by 2 equals 4. Therefore, we can simplify $\sqrt{2^2}$ as 2.

Combining the Simplified Square Roots

Now that we have simplified $\sqrt{20}$, we can rewrite the original expression as $14 \times 2 \times \sqrt{5} - 6 \sqrt{5}$.

Combining Like Terms

We can combine the like terms in the expression by factoring out the common term $\sqrt{5}$. This gives us $28 \sqrt{5} - 6 \sqrt{5}$.

Final Simplification

Now that we have combined the like terms, we can simplify the expression further by subtracting the two terms. This gives us $22 \sqrt{5}$.

Conclusion

In this article, we simplified the expression $14 \sqrt{20} - 6 \sqrt{5}$ by breaking down the square roots individually and using the properties of square roots. We started by simplifying $\sqrt{20}$ by breaking it down into its prime factors and using the property of square roots to rewrite it as $\sqrt{2^2} \times \sqrt{5}$. We then simplified $\sqrt{2^2}$ as 2 and combined the simplified square roots to get $14 \times 2 \times \sqrt{5} - 6 \sqrt{5}$. Finally, we combined the like terms and simplified the expression further to get $22 \sqrt{5}$.

Frequently Asked Questions

• What is the simplified form of $14 \sqrt{20} - 6 \sqrt{5}$? The simplified form of $14 \sqrt{20} - 6 \sqrt{5}$ is $22 \sqrt{5}$.
• How do you simplify square roots? To simplify square roots, you can break them down into their prime factors and use the property of square roots to rewrite them as the product of the square roots.
• What is the property of square roots? The property of square roots states that the square root of a product is equal to the product of the square roots.

Step-by-Step Solution

1. Simplify $\sqrt{20}$ by breaking it down into its prime factors.
2. Rewrite $\sqrt{20}$ as $\sqrt{2^2} \times \sqrt{5}$ using the property of square roots.
3. Simplify $\sqrt{2^2}$ as 2.
4. Combine the simplified square roots to get $14 \times 2 \times \sqrt{5} - 6 \sqrt{5}$.
5. Combine the like terms in the expression by factoring out the common term $\sqrt{5}$.
6. Simplify the expression further by subtracting the two terms.

Additional Resources

• Mathematics Formulas and Equations: A comprehensive list of mathematics formulas and equations, including those related to square roots.
• Algebra and Geometry: A detailed explanation of algebra and geometry, including topics such as simplifying expressions and solving equations.
• Mathematics Tutorials and Examples: A collection of mathematics tutorials and examples, including those related to simplifying expressions and solving equations.

Final Thoughts

Simplifying expressions involving square roots is a crucial skill in mathematics, particularly in algebra and geometry. By breaking down the square roots individually and using the properties of square roots, we can simplify complex expressions and arrive at a clear and concise solution. In this article, we simplified the expression $14 \sqrt{20} - 6 \sqrt{5}$ by following these steps and arrived at the final solution of $22 \sqrt{5}$.

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Introduction

In our previous article, we simplified the expression $14 \sqrt{20} - 6 \sqrt{5}$ by breaking down the square roots individually and using the properties of square roots. We arrived at the final solution of $22 \sqrt{5}$. In this article, we will answer some frequently asked questions related to simplifying expressions involving square roots.

Q&A

Q: What is the simplified form of $14 \sqrt{20} - 6 \sqrt{5}$?

A: The simplified form of $14 \sqrt{20} - 6 \sqrt{5}$ is $22 \sqrt{5}$.

Q: How do you simplify square roots?

A: To simplify square roots, you can break them down into their prime factors and use the property of square roots to rewrite them as the product of the square roots.

Q: What is the property of square roots?

A: The property of square roots states that the square root of a product is equal to the product of the square roots.

Q: Can you provide an example of simplifying a square root?

A: Yes, let's consider the square root of 18. We can break it down into its prime factors as $\sqrt{2 \times 3^2}$. Using the property of square roots, we can rewrite it as $\sqrt{2} \times \sqrt{3^2}$. Simplifying further, we get $3 \sqrt{2}$.

Q: How do you handle negative numbers when simplifying square roots?

A: When simplifying square roots, we only consider the positive square root. If the number is negative, we can rewrite it as the product of the positive square root and -1.

Q: Can you provide an example of handling negative numbers when simplifying square roots?

A: Yes, let's consider the square root of -16. We can break it down into its prime factors as $\sqrt{-1 \times 2^4}$. Using the property of square roots, we can rewrite it as $\sqrt{-1} \times \sqrt{2^4}$. Simplifying further, we get $4 \sqrt{-1}$, which can be rewritten as $4i$, where $i$ is the imaginary unit.

Q: What is the difference between a square root and a radical?

A: A square root and a radical are often used interchangeably, but technically, a radical is a more general term that refers to any root of a number, while a square root specifically refers to the root of a number that is equal to 2.

Q: Can you provide an example of a radical that is not a square root?

A: Yes, let's consider the cube root of 27. We can break it down into its prime factors as $\sqrt[3]{3^3}$. Simplifying further, we get $3$.

Conclusion

Simplifying expressions involving square roots is a crucial skill in mathematics, particularly in algebra and geometry. By breaking down the square roots individually and using the properties of square roots, we can simplify complex expressions and arrive at a clear and concise solution. In this article, we answered some frequently asked questions related to simplifying expressions involving square roots and provided examples to illustrate the concepts.

Frequently Asked Questions

• What is the simplified form of $14 \sqrt{20} - 6 \sqrt{5}$? The simplified form of $14 \sqrt{20} - 6 \sqrt{5}$ is $22 \sqrt{5}$.
• How do you simplify square roots? To simplify square roots, you can break them down into their prime factors and use the property of square roots to rewrite them as the product of the square roots.
• What is the property of square roots? The property of square roots states that the square root of a product is equal to the product of the square roots.

Step-by-Step Solution

1. Simplify the square root by breaking it down into its prime factors.
2. Rewrite the square root as the product of the square roots using the property of square roots.
3. Simplify the expression further by combining like terms.

Additional Resources

• Mathematics Formulas and Equations: A comprehensive list of mathematics formulas and equations, including those related to square roots.
• Algebra and Geometry: A detailed explanation of algebra and geometry, including topics such as simplifying expressions and solving equations.
• Mathematics Tutorials and Examples: A collection of mathematics tutorials and examples, including those related to simplifying expressions and solving equations.

Final Thoughts

Simplifying expressions involving square roots is a crucial skill in mathematics, particularly in algebra and geometry. By breaking down the square roots individually and using the properties of square roots, we can simplify complex expressions and arrive at a clear and concise solution. In this article, we answered some frequently asked questions related to simplifying expressions involving square roots and provided examples to illustrate the concepts.