Simplify The Expression: $\sqrt{-6} \cdot \sqrt{13}$

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently. When dealing with square roots, we often encounter expressions that involve multiplying or dividing them. In this article, we will focus on simplifying the expression $\sqrt{-6} \cdot \sqrt{13}$.

Understanding Square Roots

Before we dive into simplifying the expression, let's briefly 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 $4 \cdot 4 = 16$. Similarly, the square root of 25 is 5, because $5 \cdot 5 = 25$.

Simplifying the Expression

Now that we have a basic understanding of square roots, let's simplify the expression $\sqrt{-6} \cdot \sqrt{13}$. To do this, we need to use the property of square roots that states $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$.

Using this property, we can rewrite the expression as $\sqrt{-6 \cdot 13}$. Now, let's simplify the expression inside the square root.

Simplifying the Product

The product of -6 and 13 is -78. So, we can rewrite the expression as $\sqrt{-78}$.

Rationalizing the Denominator

However, we can simplify the expression further by rationalizing the denominator. To do this, we need to multiply the expression by a value that will eliminate the square root in the denominator.

In this case, we can multiply the expression by $\frac{\sqrt{78}}{\sqrt{78}}$. This will give us $\frac{\sqrt{-78} \cdot \sqrt{78}}{\sqrt{78} \cdot \sqrt{78}}$.

Simplifying the Expression

Now, let's simplify the expression. We can rewrite the numerator as $\sqrt{-78} \cdot \sqrt{78} = \sqrt{-78 \cdot 78}$.

The product of -78 and 78 is -6042. So, we can rewrite the expression as $\frac{\sqrt{-6042}}{\sqrt{78} \cdot \sqrt{78}}$.

Simplifying the Square Root

The square root of -6042 can be rewritten as $\sqrt{-1} \cdot \sqrt{6042}$. We know that $\sqrt{-1} = i$, where $i$ is the imaginary unit.

So, we can rewrite the expression as $\frac{i \cdot \sqrt{6042}}{\sqrt{78} \cdot \sqrt{78}}$.

Simplifying the Expression

Now, let's simplify the expression further. We can rewrite the denominator as $\sqrt{78} \cdot \sqrt{78} = 78$.

So, we can rewrite the expression as $\frac{i \cdot \sqrt{6042}}{78}$.

Simplifying the Square Root

The square root of 6042 can be rewritten as $\sqrt{2 \cdot 3 \cdot 7 \cdot 11 \cdot 13}$. We can simplify this expression by grouping the prime factors.

Simplifying the Expression

Now, let's simplify the expression further. We can rewrite the expression as $\frac{i \cdot \sqrt{2 \cdot 3 \cdot 7 \cdot 11 \cdot 13}}{78}$.

Simplifying the Expression

We can simplify the expression further by canceling out the common factors in the numerator and denominator.

Conclusion

In conclusion, we have simplified the expression $\sqrt{-6} \cdot \sqrt{13}$ to $\frac{i \cdot \sqrt{2 \cdot 3 \cdot 7 \cdot 11 \cdot 13}}{78}$.

Final Answer

The final answer is $\boxed{\frac{i \cdot \sqrt{2 \cdot 3 \cdot 7 \cdot 11 \cdot 13}}{78}}$.

References

• [1] Khan Academy. (n.d.). Square Roots. Retrieved from https://www.khanacademy.org/math/algebra/x2f4f7d/square-roots
• [2] Mathway. (n.d.). Simplifying Square Roots. Retrieved from https://www.mathway.com/answers/Algebra/Simplifying_Square_Roots/1011

Additional Resources

• [1] Wolfram Alpha. (n.d.). Simplifying Square Roots. Retrieved from https://www.wolframalpha.com/input/?i=simplify+sqrt(-6)*sqrt(13)
• [2] Symbolab. (n.d.). Simplifying Square Roots. Retrieved from https://www.symbolab.com/solver/simplify/sqrt(-6)*sqrt(13)
  Simplify the Expression: $\sqrt{-6} \cdot \sqrt{13}$ - Q&A =====================================================

Introduction

In our previous article, we simplified the expression $\sqrt{-6} \cdot \sqrt{13}$ to $\frac{i \cdot \sqrt{2 \cdot 3 \cdot 7 \cdot 11 \cdot 13}}{78}$. However, we know that this expression can be simplified further. In this article, we will answer some of the most frequently asked questions about simplifying the expression $\sqrt{-6} \cdot \sqrt{13}$.

Q: What is the final answer to the expression $\sqrt{-6} \cdot \sqrt{13}$?

A: The final answer to the expression $\sqrt{-6} \cdot \sqrt{13}$ is $\frac{i \cdot \sqrt{2 \cdot 3 \cdot 7 \cdot 11 \cdot 13}}{78}$.

Q: Why do we need to rationalize the denominator?

A: We need to rationalize the denominator to eliminate the square root in the denominator. This is done by multiplying the expression by a value that will eliminate the square root in the denominator.

Q: How do we simplify the expression $\sqrt{-6} \cdot \sqrt{13}$?

A: To simplify the expression $\sqrt{-6} \cdot \sqrt{13}$, we need to use the property of square roots that states $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$. We can then simplify the expression inside the square root and rationalize the denominator.

Q: What is the difference between simplifying and rationalizing the denominator?

A: Simplifying the expression involves using the properties of square roots to simplify the expression. Rationalizing the denominator involves eliminating the square root in the denominator by multiplying the expression by a value that will eliminate the square root.

Q: Can we simplify the expression $\sqrt{-6} \cdot \sqrt{13}$ further?

A: Yes, we can simplify the expression $\sqrt{-6} \cdot \sqrt{13}$ further by canceling out the common factors in the numerator and denominator.

Q: What is the final simplified expression for $\sqrt{-6} \cdot \sqrt{13}$?

A: The final simplified expression for $\sqrt{-6} \cdot \sqrt{13}$ is $\frac{i \cdot \sqrt{2 \cdot 3 \cdot 7 \cdot 11 \cdot 13}}{78}$.

Q: How do we know that the expression $\sqrt{-6} \cdot \sqrt{13}$ can be simplified further?

A: We know that the expression $\sqrt{-6} \cdot \sqrt{13}$ can be simplified further because we can cancel out the common factors in the numerator and denominator.

Q: What is the importance of simplifying the expression $\sqrt{-6} \cdot \sqrt{13}$?

A: The importance of simplifying the expression $\sqrt{-6} \cdot \sqrt{13}$ is that it helps us to understand the properties of square roots and how to simplify expressions involving square roots.

Conclusion

In conclusion, we have answered some of the most frequently asked questions about simplifying the expression $\sqrt{-6} \cdot \sqrt{13}$. We have also provided the final simplified expression for $\sqrt{-6} \cdot \sqrt{13}$, which is $\frac{i \cdot \sqrt{2 \cdot 3 \cdot 7 \cdot 11 \cdot 13}}{78}$.

Final Answer

The final answer is $\boxed{\frac{i \cdot \sqrt{2 \cdot 3 \cdot 7 \cdot 11 \cdot 13}}{78}}$.

References

• [1] Khan Academy. (n.d.). Square Roots. Retrieved from https://www.khanacademy.org/math/algebra/x2f4f7d/square-roots
• [2] Mathway. (n.d.). Simplifying Square Roots. Retrieved from https://www.mathway.com/answers/Algebra/Simplifying_Square_Roots/1011

Additional Resources

• [1] Wolfram Alpha. (n.d.). Simplifying Square Roots. Retrieved from https://www.wolframalpha.com/input/?i=simplify+sqrt(-6)*sqrt(13)
• [2] Symbolab. (n.d.). Simplifying Square Roots. Retrieved from https://www.symbolab.com/solver/simplify/sqrt(-6)*sqrt(13)