Simplify:10) − 6 27 N 3 -6 \sqrt{27 N^3} − 6 27 N 3 ​

Understanding the Problem

When dealing with square roots, it's essential to simplify the expression to make it easier to work with. In this case, we're given the expression $-6 \sqrt{27 n^3}$, and we need to simplify it. To start, let's break down the expression into its components. We have a negative sign outside the square root, and inside the square root, we have $27 n^3$. Our goal is to simplify this expression and make it more manageable.

Breaking Down the Expression

To simplify the expression, we need to focus on the part inside the square root, which is $27 n^3$. We can start by factoring the number $27$. Since $27$ is a perfect cube, we can write it as $3^3$. This gives us $27 n^3 = (3^3) n^3$. Now, we can use the property of exponents that states $(a^m)^n = a^{m \cdot n}$. Applying this property, we get $(3^3) n^3 = 3^{3 \cdot 3} n^3 = 3^9 n^3$.

Simplifying the Square Root

Now that we have factored the expression inside the square root, we can simplify it. We can rewrite the expression as $-6 \sqrt{3^9 n^3}$. Since the square root of a power is equal to the power of the square root, we can rewrite this as $-6 (3^9 n^3)^{1/2}$. Simplifying further, we get $-6 \cdot 3^{9/2} \cdot n^{3/2}$.

Simplifying the Exponents

Now that we have simplified the expression inside the square root, we can simplify the exponents. We have $-6 \cdot 3^{9/2} \cdot n^{3/2}$. To simplify the exponents, we can use the property of exponents that states $a^{m/n} = (a^m)^{1/n}$. Applying this property, we get $-6 \cdot (3^9)^{1/2} \cdot (n^3)^{1/2}$. Simplifying further, we get $-6 \cdot 3^{9/2} \cdot n^{3/2}$.

Final Simplification

Now that we have simplified the exponents, we can simplify the expression further. We have $-6 \cdot 3^{9/2} \cdot n^{3/2}$. To simplify this expression, we can use the property of exponents that states $a^m \cdot a^n = a^{m+n}$. Applying this property, we get $-6 \cdot 3^{9/2} \cdot n^{3/2} = -6 \cdot 3^{9/2 + 3/2} \cdot n^{3/2} = -6 \cdot 3^{12/2} \cdot n^{3/2} = -6 \cdot 3^6 \cdot n^{3/2}$.

Conclusion

In conclusion, we have simplified the expression $-6 \sqrt{27 n^3}$ to $-6 \cdot 3^6 \cdot n^{3/2}$. This is the final simplified form of the expression. We started by breaking down the expression into its components and then simplified the part inside the square root. We used the properties of exponents to simplify the expression further and arrived at the final simplified form.

Key Takeaways

• To simplify an expression with a square root, we need to focus on the part inside the square root.
• We can use the property of exponents that states $(a^m)^n = a^{m \cdot n}$ to simplify the expression.
• We can use the property of exponents that states $a^{m/n} = (a^m)^{1/n}$ to simplify the exponents.
• We can use the property of exponents that states $a^m \cdot a^n = a^{m+n}$ to simplify the expression further.

Practice Problems

• Simplify the expression $\sqrt{16 x^4}$.
• Simplify the expression $\sqrt{9 y^6}$.
• Simplify the expression $\sqrt{25 z^8}$.

Solutions to Practice Problems

• $\sqrt{16 x^4} = \sqrt{4^2 \cdot x^4} = 4 x^2$
• $\sqrt{9 y^6} = \sqrt{3^2 \cdot y^6} = 3 y^3$
• $\sqrt{25 z^8} = \sqrt{5^2 \cdot z^8} = 5 z^4$

Final Thoughts

Simplifying expressions with square roots can be challenging, but with the right techniques and properties of exponents, we can simplify them easily. In this article, we simplified the expression $-6 \sqrt{27 n^3}$ to $-6 \cdot 3^6 \cdot n^{3/2}$. We used the properties of exponents to simplify the expression and arrived at the final simplified form. We also provided practice problems and solutions to help you practice and reinforce your understanding of simplifying expressions with square roots.

Understanding the Problem

When dealing with square roots, it's essential to simplify the expression to make it easier to work with. In this case, we're given the expression $-6 \sqrt{27 n^3}$, and we need to simplify it. To start, let's break down the expression into its components. We have a negative sign outside the square root, and inside the square root, we have $27 n^3$. Our goal is to simplify this expression and make it more manageable.

Q&A

Q: What is the first step in simplifying the expression $-6 \sqrt{27 n^3}$?

A: The first step in simplifying the expression $-6 \sqrt{27 n^3}$ is to break down the expression into its components. We need to focus on the part inside the square root, which is $27 n^3$.

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

A: To simplify the expression inside the square root, we can factor the number $27$. Since $27$ is a perfect cube, we can write it as $3^3$. This gives us $27 n^3 = (3^3) n^3$. Now, we can use the property of exponents that states $(a^m)^n = a^{m \cdot n}$ to simplify the expression further.

Q: What is the property of exponents that we use to simplify the expression inside the square root?

A: The property of exponents that we use to simplify the expression inside the square root is $(a^m)^n = a^{m \cdot n}$. This property allows us to simplify the expression by combining the exponents.

Q: How do we simplify the exponents in the expression $-6 \cdot 3^{9/2} \cdot n^{3/2}$?

A: To simplify the exponents in the expression $-6 \cdot 3^{9/2} \cdot n^{3/2}$, we can use the property of exponents that states $a^{m/n} = (a^m)^{1/n}$. This property allows us to simplify the exponents by rewriting them as a fraction.

Q: What is the final simplified form of the expression $-6 \sqrt{27 n^3}$?

A: The final simplified form of the expression $-6 \sqrt{27 n^3}$ is $-6 \cdot 3^6 \cdot n^{3/2}$.

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

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

• Not breaking down the expression into its components
• Not using the properties of exponents to simplify the expression
• Not rewriting the exponents as a fraction
• Not simplifying the expression further by combining the exponents

Practice Problems

• Simplify the expression $\sqrt{16 x^4}$.
• Simplify the expression $\sqrt{9 y^6}$.
• Simplify the expression $\sqrt{25 z^8}$.

Solutions to Practice Problems

• $\sqrt{16 x^4} = \sqrt{4^2 \cdot x^4} = 4 x^2$
• $\sqrt{9 y^6} = \sqrt{3^2 \cdot y^6} = 3 y^3$
• $\sqrt{25 z^8} = \sqrt{5^2 \cdot z^8} = 5 z^4$

Final Thoughts

Simplifying expressions with square roots can be challenging, but with the right techniques and properties of exponents, we can simplify them easily. In this article, we simplified the expression $-6 \sqrt{27 n^3}$ to $-6 \cdot 3^6 \cdot n^{3/2}$. We used the properties of exponents to simplify the expression and arrived at the final simplified form. We also provided practice problems and solutions to help you practice and reinforce your understanding of simplifying expressions with square roots.

Additional Resources

• For more practice problems and solutions, visit our website at [insert website URL].
• For additional resources and tutorials on simplifying expressions with square roots, visit our YouTube channel at [insert YouTube channel URL].

Conclusion

In conclusion, simplifying expressions with square roots requires a clear understanding of the properties of exponents and the ability to break down the expression into its components. By following the steps outlined in this article, you can simplify expressions with square roots easily and efficiently. Remember to practice regularly and seek additional resources when needed to reinforce your understanding of this important math concept.