Simplify The Radical 245 C 3 \sqrt{245 C^3} 245 C 3 ​ .A. 5 C 49 C 5 C \sqrt{49 C} 5 C 49 C ​ B. 7 C 5 C 7 C \sqrt{5 C} 7 C 5 C ​ C. 49 C 2 5 C 2 49 C^2 \sqrt{5 C^2} 49 C 2 5 C 2 ​ D. 5 C 2 7 C 5 C^2 \sqrt{7 C} 5 C 2 7 C ​

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Understanding the Problem

The given problem involves simplifying a radical expression, specifically $\sqrt{245 c^3}$. To simplify this expression, we need to identify the largest perfect square that divides the radicand, which is the number inside the square root. This will allow us to rewrite the expression in a simpler form.

Breaking Down the Radicand

To simplify the radical, we need to break down the radicand into its prime factors. The radicand is $245 c^3$, which can be factored as follows:

$$245 c^3 = 5 \cdot 7^2 \cdot c^3$$

Identifying the Largest Perfect Square

Now that we have factored the radicand, we can identify the largest perfect square that divides it. In this case, the largest perfect square is $7^2$, which is equal to $49$.

Rewriting the Expression

Using the factored form of the radicand and the largest perfect square, we can rewrite the expression as follows:

$$\sqrt{245 c^3} = \sqrt{5 \cdot 7^2 \cdot c^3} = \sqrt{5} \cdot \sqrt{7^2} \cdot \sqrt{c^3}$$

Simplifying the Expression

Now that we have rewritten the expression, we can simplify it further by combining the square roots. Since $\sqrt{7^2} = 7$, we can rewrite the expression as follows:

$$\sqrt{245 c^3} = \sqrt{5} \cdot 7 \cdot \sqrt{c^3}$$

Final Simplification

To simplify the expression further, we need to consider the exponent of $c$ inside the square root. Since $c^3$ can be written as $c^2 \cdot c$, we can rewrite the expression as follows:

$$\sqrt{245 c^3} = \sqrt{5} \cdot 7 \cdot \sqrt{c^2 \cdot c}$$

Final Answer

Using the properties of square roots, we can simplify the expression further by combining the square roots:

$$\sqrt{245 c^3} = \sqrt{5} \cdot 7 \cdot c \cdot \sqrt{c^2}$$

Since $\sqrt{c^2} = c$, we can rewrite the expression as follows:

$$\sqrt{245 c^3} = \sqrt{5} \cdot 7 \cdot c \cdot c$$

Combining the constants, we get:

$$\sqrt{245 c^3} = 7 c \sqrt{5 c^2}$$

However, we can simplify this expression further by recognizing that $\sqrt{5 c^2} = \sqrt{5} \cdot \sqrt{c^2} = \sqrt{5} \cdot c$. Therefore, the final simplified form of the expression is:

$$\sqrt{245 c^3} = 7 c \sqrt{5 c}$$

Conclusion

In this article, we simplified the radical expression $\sqrt{245 c^3}$ by identifying the largest perfect square that divides the radicand and rewriting the expression in a simpler form. We used the properties of square roots to combine the square roots and simplify the expression further. The final simplified form of the expression is $7 c \sqrt{5 c}$.

Answer Key

The correct answer is:

• B. $7 c \sqrt{5 c}$

Additional Tips and Tricks

When simplifying radical expressions, it's essential to identify the largest perfect square that divides the radicand. This will allow you to rewrite the expression in a simpler form and combine the square roots to simplify the expression further.

Common Mistakes to Avoid

When simplifying radical expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not identifying the largest perfect square: Failing to identify the largest perfect square that divides the radicand can lead to an incorrect simplified form.
• Not combining the square roots: Failing to combine the square roots can lead to an incorrect simplified form.
• Not considering the exponent of the variable: Failing to consider the exponent of the variable inside the square root can lead to an incorrect simplified form.

Practice Problems

To practice simplifying radical expressions, try the following problems:

• $\sqrt{24 x^5}$
• $\sqrt{48 y^3}$
• $\sqrt{75 z^4}$

Conclusion

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Q: What is the largest perfect square that divides the radicand?

A: The largest perfect square that divides the radicand is $7^2$, which is equal to $49$.

Q: How do I rewrite the expression using the factored form of the radicand?

A: To rewrite the expression, we can use the factored form of the radicand, which is $5 \cdot 7^2 \cdot c^3$. We can then rewrite the expression as $\sqrt{5 \cdot 7^2 \cdot c^3} = \sqrt{5} \cdot \sqrt{7^2} \cdot \sqrt{c^3}$.

Q: How do I simplify the expression further by combining the square roots?

A: To simplify the expression further, we can combine the square roots. Since $\sqrt{7^2} = 7$, we can rewrite the expression as $\sqrt{5} \cdot 7 \cdot \sqrt{c^3}$.

Q: What is the final simplified form of the expression?

A: The final simplified form of the expression is $7 c \sqrt{5 c}$.

Q: What are some common mistakes to avoid when simplifying radical expressions?

A: Some common mistakes to avoid when simplifying radical expressions include:

• Not identifying the largest perfect square that divides the radicand
• Not combining the square roots
• Not considering the exponent of the variable inside the square root

Q: How can I practice simplifying radical expressions?

A: To practice simplifying radical expressions, try the following problems:

• $\sqrt{24 x^5}$
• $\sqrt{48 y^3}$
• $\sqrt{75 z^4}$

Q: What are some additional tips and tricks for simplifying radical expressions?

A: Some additional tips and tricks for simplifying radical expressions include:

• Identifying the largest perfect square that divides the radicand
• Combining the square roots
• Considering the exponent of the variable inside the square root

Q: How can I apply the concepts learned in this article to real-world problems?

A: The concepts learned in this article can be applied to real-world problems in various fields, including mathematics, science, and engineering. For example, simplifying radical expressions can be used to solve problems involving distance, rate, and time, as well as problems involving area and volume.

Q: What are some common applications of radical expressions in real-world problems?

A: Some common applications of radical expressions in real-world problems include:

• Calculating distances and rates
• Finding areas and volumes
• Solving problems involving physics and engineering

Q: How can I use technology to simplify radical expressions?

A: Technology can be used to simplify radical expressions by using calculators or computer software to perform calculations and simplify expressions. Additionally, online resources and tools can be used to practice and learn simplifying radical expressions.

Q: What are some online resources and tools for learning and practicing simplifying radical expressions?

A: Some online resources and tools for learning and practicing simplifying radical expressions include:

• Khan Academy
• Mathway
• Wolfram Alpha
• Online math textbooks and resources

Conclusion

In this article, we provided a Q&A section to help readers understand and apply the concepts learned in the article. We covered topics such as identifying the largest perfect square that divides the radicand, rewriting the expression using the factored form of the radicand, and simplifying the expression further by combining the square roots. We also provided additional tips and tricks for simplifying radical expressions, as well as common mistakes to avoid.