Simplify The Expression: $\sqrt{8} \times \sqrt{128}$

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Introduction

In mathematics, simplifying expressions is an essential skill that helps us solve problems efficiently and accurately. One of the most common types of expressions that require simplification is those involving square roots. In this article, we will focus on simplifying the expression $\sqrt{8} \times \sqrt{128}$.

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.

Breaking Down the Expression

Now that we have a basic understanding of square roots, let's break down the expression $\sqrt{8} \times \sqrt{128}$. We can start by simplifying each square root individually.

Simplifying $\sqrt{8}$

To simplify $\sqrt{8}$, we need to find the largest perfect square that divides 8. A perfect square is a number that can be expressed as the product of an integer with itself. For example, 4 is a perfect square because it can be expressed as 2 multiplied by 2.

The largest perfect square that divides 8 is 4. We can rewrite 8 as 4 multiplied by 2. Therefore, $\sqrt{8}$ can be simplified as $\sqrt{4 \times 2}$.

Using the property of square roots that $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$, we can rewrite $\sqrt{4 \times 2}$ as $\sqrt{4} \times \sqrt{2}$. Since $\sqrt{4}$ is equal to 2, we can simplify $\sqrt{8}$ as 2$\sqrt{2}$.

Simplifying $\sqrt{128}$

To simplify $\sqrt{128}$, we need to find the largest perfect square that divides 128. The largest perfect square that divides 128 is 64. We can rewrite 128 as 64 multiplied by 2. Therefore, $\sqrt{128}$ can be simplified as $\sqrt{64 \times 2}$.

Using the property of square roots that $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$, we can rewrite $\sqrt{64 \times 2}$ as $\sqrt{64} \times \sqrt{2}$. Since $\sqrt{64}$ is equal to 8, we can simplify $\sqrt{128}$ as 8$\sqrt{2}$.

Multiplying the Simplified Expressions

Now that we have simplified both $\sqrt{8}$ and $\sqrt{128}$, we can multiply the two expressions together.

$\sqrt{8} \times \sqrt{128} = (2\sqrt{2}) \times (8\sqrt{2})$

Using the property of square roots that $\sqrt{a} \times \sqrt{a} = a$, we can rewrite the expression as:

$(2\sqrt{2}) \times (8\sqrt{2}) = 2 \times 8 \times \sqrt{2} \times \sqrt{2}$

Since $\sqrt{2} \times \sqrt{2} = 2$, we can simplify the expression as:

$2 \times 8 \times \sqrt{2} \times \sqrt{2} = 16 \times 2$

Therefore, the final simplified expression is:

$\sqrt{8} \times \sqrt{128} = 32$

Conclusion

In this article, we simplified the expression $\sqrt{8} \times \sqrt{128}$ by breaking down each square root individually and then multiplying the simplified expressions together. We used the properties of square roots to simplify the expressions and arrived at the final answer of 32.

Tips and Tricks

• When simplifying expressions involving square roots, always look for the largest perfect square that divides the number.
• Use the property of square roots that $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ to simplify expressions.
• When multiplying expressions involving square roots, use the property of square roots that $\sqrt{a} \times \sqrt{a} = a$ to simplify the expression.

Q: What is the largest perfect square that divides 8?

A: The largest perfect square that divides 8 is 4. We can rewrite 8 as 4 multiplied by 2.

Q: How do I simplify $\sqrt{8}$?

A: To simplify $\sqrt{8}$, we need to find the largest perfect square that divides 8. We can rewrite 8 as 4 multiplied by 2. Therefore, $\sqrt{8}$ can be simplified as $\sqrt{4 \times 2}$.

Using the property of square roots that $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$, we can rewrite $\sqrt{4 \times 2}$ as $\sqrt{4} \times \sqrt{2}$. Since $\sqrt{4}$ is equal to 2, we can simplify $\sqrt{8}$ as 2$\sqrt{2}$.

Q: What is the largest perfect square that divides 128?

A: The largest perfect square that divides 128 is 64. We can rewrite 128 as 64 multiplied by 2.

Q: How do I simplify $\sqrt{128}$?

A: To simplify $\sqrt{128}$, we need to find the largest perfect square that divides 128. We can rewrite 128 as 64 multiplied by 2. Therefore, $\sqrt{128}$ can be simplified as $\sqrt{64 \times 2}$.

Using the property of square roots that $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$, we can rewrite $\sqrt{64 \times 2}$ as $\sqrt{64} \times \sqrt{2}$. Since $\sqrt{64}$ is equal to 8, we can simplify $\sqrt{128}$ as 8$\sqrt{2}$.

Q: How do I multiply expressions involving square roots?

A: When multiplying expressions involving square roots, use the property of square roots that $\sqrt{a} \times \sqrt{a} = a$ to simplify the expression.

For example, let's multiply the expressions 2$\sqrt{2}$ and 8$\sqrt{2}$.

$(2\sqrt{2}) \times (8\sqrt{2}) = 2 \times 8 \times \sqrt{2} \times \sqrt{2}$

Since $\sqrt{2} \times \sqrt{2} = 2$, we can simplify the expression as:

$2 \times 8 \times \sqrt{2} \times \sqrt{2} = 16 \times 2$

Therefore, the final simplified expression is:

$(2\sqrt{2}) \times (8\sqrt{2}) = 32$

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

A: Some common mistakes to avoid when simplifying expressions involving square roots include:

• Not finding the largest perfect square that divides the number.
• Not using the property of square roots that $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ to simplify expressions.
• Not using the property of square roots that $\sqrt{a} \times \sqrt{a} = a$ to simplify expressions.

Q: How can I practice simplifying expressions involving square roots?

A: You can practice simplifying expressions involving square roots by:

• Working through examples and exercises in your textbook or online resources.
• Creating your own examples and exercises to practice simplifying expressions involving square roots.
• Using online resources or math apps to practice simplifying expressions involving square roots.

By following these tips and practicing regularly, you can become more confident and proficient in simplifying expressions involving square roots.