What Is The Following Product?$\sqrt{30} \cdot \sqrt{10}$A. $2 \sqrt{10}$ B. $3 \sqrt{10}$ C. $4 \sqrt{10}$ D. $10 \sqrt{3}$

Understanding the Problem

The given problem involves the multiplication of two square roots, $\sqrt{30}$ and $\sqrt{10}$. To solve this problem, we need to apply the properties of square roots and simplify the expression.

Properties of Square Roots

One of the fundamental properties of square roots is that the square root of a product is equal to the product of the square roots. Mathematically, this can be expressed as:

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

This property will be useful in simplifying the given expression.

Simplifying the Expression

Using the property mentioned above, we can simplify the given expression as follows:

$$\sqrt{30} \cdot \sqrt{10} = \sqrt{30 \cdot 10}$$

Now, we need to simplify the expression inside the square root. We can do this by multiplying the numbers inside the square root:

$$\sqrt{30 \cdot 10} = \sqrt{300}$$

Simplifying the Square Root

The square root of 300 can be simplified further by factoring out the perfect square. We can write 300 as:

$$300 = 100 \cdot 3$$

Now, we can take the square root of 100, which is 10, and multiply it by the square root of 3:

$$\sqrt{300} = \sqrt{100 \cdot 3} = 10 \sqrt{3}$$

Conclusion

Therefore, the product of $\sqrt{30}$ and $\sqrt{10}$ is $10 \sqrt{3}$.

Answer

The correct answer is:

D. $10 \sqrt{3}$

Why is this the Correct Answer?

The correct answer is $10 \sqrt{3}$ because we simplified the expression using the properties of square roots and factored out the perfect square. This resulted in the final answer, which is $10 \sqrt{3}$.

Common Mistakes

Some common mistakes that students make when solving this problem include:

• Not applying the properties of square roots correctly
• Not factoring out the perfect square
• Not simplifying the expression correctly

Tips and Tricks

To avoid these common mistakes, students should:

• Make sure to apply the properties of square roots correctly
• Factor out the perfect square to simplify the expression
• Double-check their work to ensure that the expression is simplified correctly

Real-World Applications

The concept of square roots and their properties has many real-world applications, including:

• Physics: Square roots are used to calculate distances and velocities in physics.
• Engineering: Square roots are used to calculate stresses and strains in engineering.
• Computer Science: Square roots are used in algorithms for solving problems in computer science.

Conclusion

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Frequently Asked Questions

We have received many questions about the problem of finding the product of $\sqrt{30}$ and $\sqrt{10}$. Here are some of the most frequently asked questions and their answers:

Q: What is the property of square roots that is used to simplify the expression?

A: The property of square roots that is used to simplify the expression is that the square root of a product is equal to the product of the square roots. Mathematically, this can be expressed as:

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

Q: How do I simplify the expression inside the square root?

A: To simplify the expression inside the square root, you need to factor out the perfect square. In this case, we can write 300 as:

$$300 = 100 \cdot 3$$

Now, we can take the square root of 100, which is 10, and multiply it by the square root of 3:

$$\sqrt{300} = \sqrt{100 \cdot 3} = 10 \sqrt{3}$$

Q: Why is the answer $10 \sqrt{3}$ and not $2 \sqrt{10}$ or $3 \sqrt{10}$?

A: The answer $10 \sqrt{3}$ is correct because we simplified the expression using the properties of square roots and factored out the perfect square. The other options, $2 \sqrt{10}$ and $3 \sqrt{10}$, are not correct because they do not take into account the properties of square roots and the factoring of the perfect square.

Q: What are some common mistakes that students make when solving this problem?

A: Some common mistakes that students make when solving this problem include:

• Not applying the properties of square roots correctly
• Not factoring out the perfect square
• Not simplifying the expression correctly

Q: How can I avoid making these common mistakes?

A: To avoid making these common mistakes, you should:

• Make sure to apply the properties of square roots correctly
• Factor out the perfect square to simplify the expression
• Double-check your work to ensure that the expression is simplified correctly

Q: What are some real-world applications of the concept of square roots?

A: The concept of square roots and their properties has many real-world applications, including:

• Physics: Square roots are used to calculate distances and velocities in physics.
• Engineering: Square roots are used to calculate stresses and strains in engineering.
• Computer Science: Square roots are used in algorithms for solving problems in computer science.

Q: Can you provide more examples of problems that involve the product of square roots?

A: Yes, here are a few more examples of problems that involve the product of square roots:

• $\sqrt{24} \cdot \sqrt{9} = ?$
• $\sqrt{36} \cdot \sqrt{16} = ?$
• $\sqrt{49} \cdot \sqrt{25} = ?$

These problems can be solved using the same properties of square roots and simplification techniques that we used to solve the original problem.

Conclusion

In conclusion, the product of $\sqrt{30}$ and $\sqrt{10}$ is $10 \sqrt{3}$. This problem requires the application of the properties of square roots and simplification of the expression. By following the correct steps and avoiding common mistakes, students can arrive at the correct answer. We hope that this Q&A article has been helpful in clarifying any questions or doubts that you may have had about this problem.