Simplify 63 W 5 \sqrt{63 W^5} 63 W 5 ​ .Assume That The Variable Represents A Positive Real Number.

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Understanding the Problem

When simplifying the given expression, $\sqrt{63 w^5}$, we need to assume that the variable $w$ represents a positive real number. This is crucial because the square root of a negative number is not a real number, and we are dealing with a real number here.

Breaking Down the Expression

To simplify the expression, we can break it down into two parts: the square root of $63$ and the square root of $w^5$. We can simplify the square root of $63$ by finding its prime factors.

Simplifying the Square Root of 63

The prime factorization of $63$ is $3^2 \times 7$. Therefore, we can rewrite the square root of $63$ as $\sqrt{3^2 \times 7}$.

Applying the Property of Square Roots

Using the property of square roots, we can rewrite $\sqrt{3^2 \times 7}$ as $\sqrt{3^2} \times \sqrt{7}$. This simplifies to $3\sqrt{7}$.

Simplifying the Square Root of $w^5$

Now, let's simplify the square root of $w^5$. We can rewrite $w^5$ as $w^4 \times w$. Using the property of square roots, we can rewrite $\sqrt{w^4 \times w}$ as $\sqrt{w^4} \times \sqrt{w}$.

Applying the Property of Square Roots Again

Using the property of square roots, we can rewrite $\sqrt{w^4}$ as $w^2$. Therefore, we can rewrite $\sqrt{w^5}$ as $w^2\sqrt{w}$.

Combining the Simplified Expressions

Now, let's combine the simplified expressions for the square root of $63$ and the square root of $w^5$. We have $3\sqrt{7} \times w^2\sqrt{w}$.

Final Simplification

Using the property of square roots, we can rewrite $3\sqrt{7} \times w^2\sqrt{w}$ as $3w^2\sqrt{7w}$.

Conclusion

In conclusion, the simplified expression for $\sqrt{63 w^5}$ is $3w^2\sqrt{7w}$.

Example Use Case

Suppose we have a problem where we need to find the value of $\sqrt{63 w^5}$, where $w$ is a positive real number. We can use the simplified expression $3w^2\sqrt{7w}$ to find the value.

Step-by-Step Solution

Here's a step-by-step solution to the problem:

1. Break down the expression into two parts: the square root of $63$ and the square root of $w^5$.
2. Simplify the square root of $63$ by finding its prime factors.
3. Apply the property of square roots to simplify the square root of $63$.
4. Simplify the square root of $w^5$ by rewriting it as $w^4 \times w$.
5. Apply the property of square roots to simplify the square root of $w^5$.
6. Combine the simplified expressions for the square root of $63$ and the square root of $w^5$.
7. Use the property of square roots to rewrite the combined expression.

Final Answer

The final answer is $3w^2\sqrt{7w}$.

Related Problems

Here are some related problems that you can try:

• Simplify $\sqrt{48 x^7}$.
• Simplify $\sqrt{75 y^3}$.
• Simplify $\sqrt{27 z^9}$.

Practice Problems

Here are some practice problems that you can try:

• Simplify $\sqrt{16 a^4}$.
• Simplify $\sqrt{25 b^6}$.
• Simplify $\sqrt{36 c^8}$.

Conclusion

In conclusion, simplifying the expression $\sqrt{63 w^5}$ involves breaking it down into two parts, simplifying the square root of $63$, and simplifying the square root of $w^5$. By applying the property of square roots, we can rewrite the expression as $3w^2\sqrt{7w}$. This is a useful technique to have in your toolkit when dealing with square roots and exponents.

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Frequently Asked Questions

Q: What is the simplified expression for $\sqrt{63 w^5}$?

A: The simplified expression for $\sqrt{63 w^5}$ is $3w^2\sqrt{7w}$.

Q: How do I simplify the square root of $63$?

A: To simplify the square root of $63$, you need to find its prime factors. The prime factorization of $63$ is $3^2 \times 7$. Therefore, you can rewrite the square root of $63$ as $\sqrt{3^2 \times 7}$.

Q: How do I simplify the square root of $w^5$?

A: To simplify the square root of $w^5$, you can rewrite it as $w^4 \times w$. Using the property of square roots, you can rewrite $\sqrt{w^4 \times w}$ as $\sqrt{w^4} \times \sqrt{w}$.

Q: What is the property of square roots that I can use to simplify the expression?

A: The property of square roots that you can use to simplify the expression is $\sqrt{a^2} = a$ and $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$.

Q: How do I combine the simplified expressions for the square root of $63$ and the square root of $w^5$?

A: To combine the simplified expressions, you can multiply them together. Therefore, you have $3\sqrt{7} \times w^2\sqrt{w}$.

Q: What is the final simplified expression for $\sqrt{63 w^5}$?

A: The final simplified expression for $\sqrt{63 w^5}$ is $3w^2\sqrt{7w}$.

Q: Can I use this technique to simplify other expressions with square roots and exponents?

A: Yes, you can use this technique to simplify other expressions with square roots and exponents. Just remember to break down the expression into two parts, simplify each part separately, and then combine them.

Q: What are some related problems that I can try?

A: Here are some related problems that you can try:

• Simplify $\sqrt{48 x^7}$.
• Simplify $\sqrt{75 y^3}$.
• Simplify $\sqrt{27 z^9}$.

Q: What are some practice problems that I can try?

A: Here are some practice problems that you can try:

• Simplify $\sqrt{16 a^4}$.
• Simplify $\sqrt{25 b^6}$.
• Simplify $\sqrt{36 c^8}$.

Additional Resources

If you need more help or want to learn more about simplifying expressions with square roots and exponents, here are some additional resources that you can use:

• Khan Academy: Simplifying Square Roots
• Mathway: Simplifying Square Roots
• Wolfram Alpha: Simplifying Square Roots

Conclusion

In conclusion, simplifying the expression $\sqrt{63 w^5}$ involves breaking it down into two parts, simplifying the square root of $63$, and simplifying the square root of $w^5$. By applying the property of square roots, we can rewrite the expression as $3w^2\sqrt{7w}$. This is a useful technique to have in your toolkit when dealing with square roots and exponents.