Simplify: $\sqrt{16 N^2}$.Provide Your Answer Below:

Understanding the Problem

When simplifying the given expression, $\sqrt{16 n^2}$, we need to apply the properties of radicals and exponents to simplify it. The expression involves a square root of a product of two terms, $16$ and $n^2$. Our goal is to simplify this expression and provide a final answer.

Breaking Down the Expression

To simplify the expression, we can start by breaking it down into its components. The expression can be rewritten as $\sqrt{16} \cdot \sqrt{n^2}$. This allows us to separate the square root of the product into the product of the square roots.

Simplifying the Square Root of 16

The square root of $16$ can be simplified as $\sqrt{16} = 4$. This is because $4^2 = 16$, and the square root of a perfect square is the number that, when multiplied by itself, gives the original number.

Simplifying the Square Root of $n^2$

The square root of $n^2$ can be simplified as $\sqrt{n^2} = n$. This is because the square root of a perfect square is the number that, when multiplied by itself, gives the original number.

Combining the Simplified Expressions

Now that we have simplified the individual components, we can combine them to get the final simplified expression. The expression $\sqrt{16 n^2}$ can be rewritten as $4n$. This is because we multiplied the simplified expressions $\sqrt{16}$ and $\sqrt{n^2}$ together.

Conclusion

In conclusion, the simplified expression for $\sqrt{16 n^2}$ is $4n$. This is achieved by breaking down the expression into its components, simplifying the square root of $16$ and $n^2$, and then combining the simplified expressions.

Final Answer

The final answer is: $\boxed{4n}$

Understanding the Problem

When simplifying the given expression, $\sqrt{16 n^2}$, we need to apply the properties of radicals and exponents to simplify it. The expression involves a square root of a product of two terms, $16$ and $n^2$. Our goal is to simplify this expression and provide a final answer.

Breaking Down the Expression

To simplify the expression, we can start by breaking it down into its components. The expression can be rewritten as $\sqrt{16} \cdot \sqrt{n^2}$. This allows us to separate the square root of the product into the product of the square roots.

Simplifying the Square Root of 16

The square root of $16$ can be simplified as $\sqrt{16} = 4$. This is because $4^2 = 16$, and the square root of a perfect square is the number that, when multiplied by itself, gives the original number.

Simplifying the Square Root of $n^2$

The square root of $n^2$ can be simplified as $\sqrt{n^2} = n$. This is because the square root of a perfect square is the number that, when multiplied by itself, gives the original number.

Combining the Simplified Expressions

Now that we have simplified the individual components, we can combine them to get the final simplified expression. The expression $\sqrt{16 n^2}$ can be rewritten as $4n$. This is because we multiplied the simplified expressions $\sqrt{16}$ and $\sqrt{n^2}$ together.

Q&A

Q: What is the simplified expression for $\sqrt{16 n^2}$?

A: The simplified expression for $\sqrt{16 n^2}$ is $4n$.

Q: How do you simplify the square root of $16$?

A: The square root of $16$ can be simplified as $\sqrt{16} = 4$. This is because $4^2 = 16$, and the square root of a perfect square is the number that, when multiplied by itself, gives the original number.

Q: How do you simplify the square root of $n^2$?

A: The square root of $n^2$ can be simplified as $\sqrt{n^2} = n$. This is because the square root of a perfect square is the number that, when multiplied by itself, gives the original number.

Q: What is the property of radicals that allows us to separate the square root of a product into the product of the square roots?

A: The property of radicals that allows us to separate the square root of a product into the product of the square roots is the product rule of radicals, which states that $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$.

Q: Can you provide an example of how to apply the product rule of radicals?

A: Yes, the expression $\sqrt{16 n^2}$ can be rewritten as $\sqrt{16} \cdot \sqrt{n^2}$. This allows us to separate the square root of the product into the product of the square roots.

Conclusion

In conclusion, the simplified expression for $\sqrt{16 n^2}$ is $4n$. This is achieved by breaking down the expression into its components, simplifying the square root of $16$ and $n^2$, and then combining the simplified expressions.

Final Answer

The final answer is: $\boxed{4n}$