Simplify The Expression: ${ 8 \sqrt{40} + \sqrt{160} }$

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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: $8 \sqrt{40} + \sqrt{160}$. We will break down the process step by step, using various techniques to simplify the expression.

Understanding the Expression

The given expression consists of two terms: $8 \sqrt{40}$ and $\sqrt{160}$. To simplify the expression, we need to understand the properties of square roots and how to manipulate them.

Properties of Square Roots

• The square root of a number is a value that, when multiplied by itself, gives the original number.
• The square root of a product is equal to the product of the square roots: $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$.
• The square root of a quotient is equal to the quotient of the square roots: $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$.

Simplifying the First Term

The first term is $8 \sqrt{40}$. To simplify this term, we can start by factoring the number inside the square root.

Factoring the Number Inside the Square Root

The number inside the square root is $40$. We can factor $40$ as follows:

$40 = 4 \cdot 10$

Now, we can rewrite the first term as:

$8 \sqrt{40} = 8 \sqrt{4 \cdot 10}$

Using the property of square roots mentioned earlier, we can simplify the expression as follows:

$8 \sqrt{4 \cdot 10} = 8 \cdot \sqrt{4} \cdot \sqrt{10}$

Simplifying the Square Root of 4

The square root of $4$ is $2$, so we can simplify the expression as follows:

$8 \cdot \sqrt{4} \cdot \sqrt{10} = 8 \cdot 2 \cdot \sqrt{10}$

Simplifying the First Term

Now, we can simplify the first term as follows:

$8 \cdot 2 \cdot \sqrt{10} = 16 \sqrt{10}$

Simplifying the Second Term

The second term is $\sqrt{160}$. To simplify this term, we can start by factoring the number inside the square root.

Factoring the Number Inside the Square Root

The number inside the square root is $160$. We can factor $160$ as follows:

$160 = 16 \cdot 10$

Now, we can rewrite the second term as:

$\sqrt{160} = \sqrt{16 \cdot 10}$

Using the property of square roots mentioned earlier, we can simplify the expression as follows:

$\sqrt{16 \cdot 10} = \sqrt{16} \cdot \sqrt{10}$

Simplifying the Square Root of 16

The square root of $16$ is $4$, so we can simplify the expression as follows:

$\sqrt{16} \cdot \sqrt{10} = 4 \cdot \sqrt{10}$

Simplifying the Second Term

Now, we can simplify the second term as follows:

$4 \cdot \sqrt{10} = 4 \sqrt{10}$

Combining the Terms

Now that we have simplified both terms, we can combine them to get the final expression.

Combining the Terms

The final expression is:

$16 \sqrt{10} + 4 \sqrt{10}$

We can combine the terms by adding the coefficients:

$16 \sqrt{10} + 4 \sqrt{10} = (16 + 4) \sqrt{10}$

Simplifying the Final Expression

Now, we can simplify the final expression as follows:

$(16 + 4) \sqrt{10} = 20 \sqrt{10}$

Conclusion

In this article, we simplified the expression $8 \sqrt{40} + \sqrt{160}$ using various techniques. We factored the numbers inside the square roots, used the properties of square roots, and combined the terms to get the final expression. The final expression is $20 \sqrt{10}$.

Frequently Asked Questions

• What is the square root of 40?
• What is the square root of 160?
• How do you simplify an expression involving square roots?
• What are the properties of square roots?

Final Answer

The final answer is: $\boxed{20 \sqrt{10}}$

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Introduction

In our previous article, we simplified the expression $8 \sqrt{40} + \sqrt{160}$ using various techniques. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q: What is the square root of 40?

A: The square root of 40 is $\sqrt{40} = \sqrt{4 \cdot 10} = 2 \sqrt{10}$.

Q: What is the square root of 160?

A: The square root of 160 is $\sqrt{160} = \sqrt{16 \cdot 10} = 4 \sqrt{10}$.

Q: How do you simplify an expression involving square roots?

A: To simplify an expression involving square roots, you can use the following steps:

1. Factor the numbers inside the square roots.
2. Use the properties of square roots, such as $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ and $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$.
3. Combine the terms to get the final expression.

Q: What are the properties of square roots?

A: The properties of square roots are:

• $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$
• $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$
• $\sqrt{a^2} = a$

Q: How do you simplify a square root of a product?

A: To simplify a square root of a product, you can use the property $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$. For example, $\sqrt{12} = \sqrt{4 \cdot 3} = 2 \sqrt{3}$.

Q: How do you simplify a square root of a quotient?

A: To simplify a square root of a quotient, you can use the property $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$. For example, $\sqrt{\frac{9}{16}} = \frac{\sqrt{9}}{\sqrt{16}} = \frac{3}{4}$.

Q: How do you simplify a square root of a perfect square?

A: To simplify a square root of a perfect square, you can use the property $\sqrt{a^2} = a$. For example, $\sqrt{16} = \sqrt{4^2} = 4$.

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions involving square roots. We covered topics such as the square root of 40, the square root of 160, and the properties of square roots. We also provided examples of how to simplify square roots of products, quotients, and perfect squares.

Final Answer

The final answer is: $\boxed{20 \sqrt{10}}$

Additional Resources

• Simplifying Expressions Involving Square Roots
• Properties of Square Roots
• Simplifying Square Roots of Products
• Simplifying Square Roots of Quotients
• Simplifying Square Roots of Perfect Squares