Simplify: 2 ⋅ 7 \sqrt{2} \cdot \sqrt{7} 2 ​ ⋅ 7 ​

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Understanding the Problem

When dealing with square roots, it's essential to understand the properties and rules that govern their behavior. In this case, we're tasked with simplifying the expression $\sqrt{2} \cdot \sqrt{7}$. To approach this problem, we need to recall the rule that states the product of two square roots is equal to the square root of the product of the numbers inside the square roots.

The Rule for Multiplying Square Roots

The rule for multiplying square roots is as follows:

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

This rule allows us to simplify expressions involving square roots by combining the numbers inside the square roots.

Applying the Rule to the Given Expression

Now that we've recalled the rule for multiplying square roots, we can apply it to the given expression:

$$\sqrt{2} \cdot \sqrt{7} = \sqrt{2 \cdot 7}$$

Using the rule, we can simplify the expression by combining the numbers inside the square roots.

Simplifying the Expression

To simplify the expression, we need to multiply the numbers inside the square roots:

$$\sqrt{2 \cdot 7} = \sqrt{14}$$

Therefore, the simplified expression is $\sqrt{14}$.

Conclusion

In this article, we've simplified the expression $\sqrt{2} \cdot \sqrt{7}$ using the rule for multiplying square roots. By applying this rule, we were able to combine the numbers inside the square roots and simplify the expression to $\sqrt{14}$. This demonstrates the importance of understanding the properties and rules that govern square roots in mathematics.

Real-World Applications

The concept of multiplying square roots has numerous real-world applications in various fields, including physics, engineering, and computer science. For example, in physics, the concept of wave-particle duality is often described using square roots. In engineering, the concept of signal processing involves the use of square roots to analyze and manipulate signals. In computer science, the concept of numerical analysis involves the use of square roots to solve equations and optimize algorithms.

Common Mistakes to Avoid

When working with square roots, it's essential to avoid common mistakes that can lead to incorrect results. Some common mistakes to avoid include:

• Not applying the rule for multiplying square roots: Failing to apply the rule for multiplying square roots can lead to incorrect results.
• Not simplifying the expression: Failing to simplify the expression can lead to unnecessary complexity and errors.
• Not checking the result: Failing to check the result can lead to incorrect conclusions and mistakes.

Tips and Tricks

When working with square roots, here are some tips and tricks to keep in mind:

• Use the rule for multiplying square roots: The rule for multiplying square roots is a powerful tool for simplifying expressions involving square roots.
• Simplify the expression: Simplifying the expression can help to avoid unnecessary complexity and errors.
• Check the result: Checking the result can help to ensure that the expression is correct and accurate.

Conclusion

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Frequently Asked Questions

In this article, we'll address some of the most frequently asked questions related to simplifying the expression $\sqrt{2} \cdot \sqrt{7}$.

Q: What is the rule for multiplying square roots?

A: The rule for multiplying square roots is as follows:

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

This rule allows us to simplify expressions involving square roots by combining the numbers inside the square roots.

Q: How do I apply the rule for multiplying square roots?

A: To apply the rule for multiplying square roots, simply multiply the numbers inside the square roots and take the square root of the result. For example:

$$\sqrt{2} \cdot \sqrt{7} = \sqrt{2 \cdot 7} = \sqrt{14}$$

Q: What if the numbers inside the square roots are not integers?

A: If the numbers inside the square roots are not integers, you can still apply the rule for multiplying square roots. For example:

$$\sqrt{2} \cdot \sqrt{3.5} = \sqrt{2 \cdot 3.5} = \sqrt{7}$$

Q: Can I simplify an expression with multiple square roots?

A: Yes, you can simplify an expression with multiple square roots by applying the rule for multiplying square roots multiple times. For example:

$$\sqrt{2} \cdot \sqrt{3} \cdot \sqrt{4} = \sqrt{2 \cdot 3 \cdot 4} = \sqrt{24}$$

Q: What if I have a negative number inside a square root?

A: If you have a negative number inside a square root, you cannot simplify the expression using the rule for multiplying square roots. For example:

$$\sqrt{-2} \cdot \sqrt{3}$$

In this case, you would need to use a different method to simplify the expression.

Q: Can I simplify an expression with a square root and a number?

A: Yes, you can simplify an expression with a square root and a number by applying the rule for multiplying square roots. For example:

$$\sqrt{2} \cdot 3 = \sqrt{2 \cdot 3^2} = \sqrt{18}$$

Q: What if I have a decimal number inside a square root?

A: If you have a decimal number inside a square root, you can still apply the rule for multiplying square roots. For example:

$$\sqrt{2.5} \cdot \sqrt{3} = \sqrt{2.5 \cdot 3} = \sqrt{7.5}$$

Conclusion

In conclusion, simplifying the expression $\sqrt{2} \cdot \sqrt{7}$ using the rule for multiplying square roots is a straightforward process that involves combining the numbers inside the square roots. By applying this rule, we can simplify expressions involving square roots and arrive at the correct result. We hope this Q&A article has been helpful in addressing some of the most frequently asked questions related to simplifying the expression $\sqrt{2} \cdot \sqrt{7}$.