Simplify The Expression: 2 ⋅ 50 \sqrt{2} \cdot \sqrt{50} 2 ​ ⋅ 50 ​

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Introduction

When dealing with square roots, it's essential to understand the properties of radicals and how to simplify expressions involving them. In this article, we will focus on simplifying the expression $\sqrt{2} \cdot \sqrt{50}$ using the properties of radicals. We will break down the process step by step, explaining each concept and providing examples to help solidify the understanding.

Understanding Radicals

A radical is a mathematical expression that represents a number that can be expressed as the product of a perfect square and another number. The radical symbol, $\sqrt{}$, is used to denote the square root of a number. For example, $\sqrt{16}$ represents the number that, when multiplied by itself, gives 16. In this case, $\sqrt{16} = 4$ because $4 \cdot 4 = 16$.

Simplifying the Expression

To simplify the expression $\sqrt{2} \cdot \sqrt{50}$, we need to understand the properties of radicals. One of the key properties is that the product of two square roots is equal to the square root of the product of the numbers inside the radicals. This can be expressed as:

$$\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$$

Using this property, we can simplify the expression $\sqrt{2} \cdot \sqrt{50}$ as follows:

$$\sqrt{2} \cdot \sqrt{50} = \sqrt{2 \cdot 50}$$

Breaking Down the Product

Now that we have simplified the expression to $\sqrt{2 \cdot 50}$, we need to break down the product inside the radical. We can do this by factoring the number 50 into its prime factors:

$$50 = 2 \cdot 25$$

Simplifying the Radical

Now that we have factored the number 50, we can simplify the radical by combining the factors:

$$\sqrt{2 \cdot 50} = \sqrt{2 \cdot 2 \cdot 25}$$

Canceling Out Perfect Squares

We can simplify the radical further by canceling out the perfect squares. In this case, we have two factors of 2, which can be combined to form a perfect square:

$$\sqrt{2 \cdot 2 \cdot 25} = \sqrt{(2 \cdot 2) \cdot 25}$$

Simplifying the Expression

Now that we have canceled out the perfect squares, we can simplify the expression:

$$\sqrt{(2 \cdot 2) \cdot 25} = \sqrt{4 \cdot 25}$$

Final Simplification

Finally, we can simplify the expression by combining the factors:

$$\sqrt{4 \cdot 25} = \sqrt{4} \cdot \sqrt{25}$$

Conclusion

In this article, we simplified the expression $\sqrt{2} \cdot \sqrt{50}$ using the properties of radicals. We broke down the product inside the radical, factored the number 50, and canceled out the perfect squares to arrive at the final simplified expression. This process demonstrates the importance of understanding the properties of radicals and how to apply them to simplify complex expressions.

Additional Examples

Here are a few additional examples of simplifying expressions involving radicals:

• $\sqrt{3} \cdot \sqrt{27} = \sqrt{3 \cdot 27} = \sqrt{81} = 9$
• $\sqrt{5} \cdot \sqrt{20} = \sqrt{5 \cdot 20} = \sqrt{100} = 10$
• $\sqrt{7} \cdot \sqrt{49} = \sqrt{7 \cdot 49} = \sqrt{343} = 7\sqrt{7}$

These examples demonstrate the application of the properties of radicals to simplify complex expressions.

Final Thoughts

Simplifying expressions involving radicals requires a deep understanding of the properties of radicals and how to apply them to complex expressions. By breaking down the product inside the radical, factoring the numbers, and canceling out perfect squares, we can arrive at the final simplified expression. This process is essential in mathematics and has numerous applications in various fields, including physics, engineering, and computer science.

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Introduction

In our previous article, we simplified the expression $\sqrt{2} \cdot \sqrt{50}$ using the properties of radicals. In this article, we will provide a Q&A section to help clarify any doubts and provide additional examples to reinforce the understanding of simplifying expressions involving radicals.

Q&A

Q: What is the property of radicals that allows us to simplify the expression $\sqrt{2} \cdot \sqrt{50}$?

A: The property of radicals that allows us to simplify the expression $\sqrt{2} \cdot \sqrt{50}$ is that the product of two square roots is equal to the square root of the product of the numbers inside the radicals. This can be expressed as:

$$\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$$

Q: How do we break down the product inside the radical?

A: To break down the product inside the radical, we need to factor the numbers inside the radicals. In the case of $\sqrt{2} \cdot \sqrt{50}$, we can factor 50 as:

$$50 = 2 \cdot 25$$

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 a whole number and itself. For example, 4 is a perfect square because it can be expressed as $2 \cdot 2$. A non-perfect square is a number that cannot be expressed as the product of a whole number and itself.

Q: How do we simplify the radical by canceling out perfect squares?

A: To simplify the radical by canceling out perfect squares, we need to identify the perfect squares inside the radical and cancel them out. In the case of $\sqrt{2 \cdot 50}$, we can cancel out the perfect square 4 as follows:

$$\sqrt{2 \cdot 50} = \sqrt{2 \cdot 2 \cdot 25} = \sqrt{(2 \cdot 2) \cdot 25} = \sqrt{4 \cdot 25}$$

Q: What is the final simplified expression for $\sqrt{2} \cdot \sqrt{50}$?

A: The final simplified expression for $\sqrt{2} \cdot \sqrt{50}$ is:

$$\sqrt{2} \cdot \sqrt{50} = \sqrt{2 \cdot 50} = \sqrt{2 \cdot 2 \cdot 25} = \sqrt{(2 \cdot 2) \cdot 25} = \sqrt{4 \cdot 25} = \sqrt{4} \cdot \sqrt{25} = 2 \cdot 5 = 10$$

Additional Examples

Here are a few additional examples of simplifying expressions involving radicals:

• $\sqrt{3} \cdot \sqrt{27} = \sqrt{3 \cdot 27} = \sqrt{81} = 9$
• $\sqrt{5} \cdot \sqrt{20} = \sqrt{5 \cdot 20} = \sqrt{100} = 10$
• $\sqrt{7} \cdot \sqrt{49} = \sqrt{7 \cdot 49} = \sqrt{343} = 7\sqrt{7}$

Conclusion

In this article, we provided a Q&A section to help clarify any doubts and provide additional examples to reinforce the understanding of simplifying expressions involving radicals. We hope that this article has been helpful in understanding the properties of radicals and how to apply them to simplify complex expressions.

Final Thoughts

Simplifying expressions involving radicals requires a deep understanding of the properties of radicals and how to apply them to complex expressions. By breaking down the product inside the radical, factoring the numbers, and canceling out perfect squares, we can arrive at the final simplified expression. This process is essential in mathematics and has numerous applications in various fields, including physics, engineering, and computer science.