What Is The Following Difference?${ 11 \sqrt{45} - 4 \sqrt{5} }$A. { 7 \sqrt{40} $}$B. { 14 \sqrt{10} $}$C. { 29 \sqrt{5} $}$D. { 95 \sqrt{5} $}$

In mathematics, we often come across problems that involve simplifying expressions with square roots. One such problem is the following difference: $11 \sqrt{45} - 4 \sqrt{5}$. In this article, we will explore the solution to this problem and understand the underlying concepts.

Understanding Square Roots

Before we dive into the solution, let's quickly 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 multiplied by 4 equals 16. We can represent square roots using the symbol $\sqrt{}$. For instance, $\sqrt{16}$ represents the square root of 16.

Simplifying Square Roots

Now, let's simplify the expression $\sqrt{45}$. We can break down 45 into its prime factors: $45 = 3 \times 3 \times 5$. Since the square root of a product is equal to the product of the square roots, we can rewrite $\sqrt{45}$ as $\sqrt{3 \times 3 \times 5}$, which simplifies to $3\sqrt{5}$.

Simplifying the Expression

Now that we have simplified $\sqrt{45}$, let's simplify the original expression: $11 \sqrt{45} - 4 \sqrt{5}$. We can substitute $\sqrt{45}$ with $3\sqrt{5}$, giving us $11 \times 3\sqrt{5} - 4 \sqrt{5}$. This simplifies to $33\sqrt{5} - 4\sqrt{5}$.

Combining Like Terms

We can combine the like terms $33\sqrt{5}$ and $-4\sqrt{5}$ by adding their coefficients. This gives us $(33 - 4)\sqrt{5}$, which simplifies to $29\sqrt{5}$.

Conclusion

In conclusion, the following difference: $11 \sqrt{45} - 4 \sqrt{5}$ simplifies to $29\sqrt{5}$. This problem required us to simplify square roots and combine like terms. By understanding the underlying concepts and applying the necessary mathematical operations, we were able to arrive at the correct solution.

Answer

The correct answer is:

• C. $29 \sqrt{5}$

Why is this the correct answer?

This is the correct answer because we simplified the expression $11 \sqrt{45} - 4 \sqrt{5}$ by substituting $\sqrt{45}$ with $3\sqrt{5}$, and then combining like terms. This resulted in the simplified expression $29\sqrt{5}$, which matches the correct answer.

What is the significance of this problem?

This problem is significant because it requires us to apply mathematical operations, such as simplifying square roots and combining like terms. By solving this problem, we can develop our problem-solving skills and improve our understanding of mathematical concepts.

What are some real-world applications of this problem?

This problem has real-world applications in various fields, such as engineering, physics, and computer science. For instance, in engineering, we may need to calculate the square root of a number to determine the length of a side of a triangle. In physics, we may need to calculate the square root of a number to determine the velocity of an object. In computer science, we may need to calculate the square root of a number to determine the distance between two points.

What are some common mistakes to avoid when solving this problem?

When solving this problem, some common mistakes to avoid include:

• Not simplifying the square root of 45
• Not combining like terms
• Not checking the units of the answer

What are some tips for solving this problem?

Some tips for solving this problem include:

• Simplifying the square root of 45
• Combining like terms
• Checking the units of the answer

What are some related problems to this problem?

Some related problems to this problem include:

• Simplifying expressions with square roots
• Combining like terms
• Checking the units of the answer

What are some resources for learning more about this problem?

Some resources for learning more about this problem include:

• Online math tutorials
• Math textbooks
• Online math communities

Conclusion

$$$$

In our previous article, we explored the solution to the problem: $11 \sqrt{45} - 4 \sqrt{5}$. In this article, we will answer some frequently asked questions related to this problem.

Q: What is the square root of 45?

A: The square root of 45 can be simplified as $3\sqrt{5}$.

Q: How do I simplify the expression $11 \sqrt{45} - 4 \sqrt{5}$?

A: To simplify the expression $11 \sqrt{45} - 4 \sqrt{5}$, you can substitute $\sqrt{45}$ with $3\sqrt{5}$, giving you $11 \times 3\sqrt{5} - 4 \sqrt{5}$. This simplifies to $33\sqrt{5} - 4\sqrt{5}$.

Q: How do I combine like terms in the expression $33\sqrt{5} - 4\sqrt{5}$?

A: To combine like terms in the expression $33\sqrt{5} - 4\sqrt{5}$, you can add the coefficients of the like terms. This gives you $(33 - 4)\sqrt{5}$, which simplifies to $29\sqrt{5}$.

Q: What is the significance of this problem?

A: This problem is significant because it requires us to apply mathematical operations, such as simplifying square roots and combining like terms. By solving this problem, we can develop our problem-solving skills and improve our understanding of mathematical concepts.

Q: What are some real-world applications of this problem?

A: This problem has real-world applications in various fields, such as engineering, physics, and computer science. For instance, in engineering, we may need to calculate the square root of a number to determine the length of a side of a triangle. In physics, we may need to calculate the square root of a number to determine the velocity of an object. In computer science, we may need to calculate the square root of a number to determine the distance between two points.

Q: What are some common mistakes to avoid when solving this problem?

A: Some common mistakes to avoid when solving this problem include:

• Not simplifying the square root of 45
• Not combining like terms
• Not checking the units of the answer

Q: What are some tips for solving this problem?

A: Some tips for solving this problem include:

• Simplifying the square root of 45
• Combining like terms
• Checking the units of the answer

Q: What are some related problems to this problem?

A: Some related problems to this problem include:

• Simplifying expressions with square roots
• Combining like terms
• Checking the units of the answer

Q: What are some resources for learning more about this problem?

A: Some resources for learning more about this problem include:

• Online math tutorials
• Math textbooks
• Online math communities

Conclusion

In conclusion, the following difference: $11 \sqrt{45} - 4 \sqrt{5}$ simplifies to $29\sqrt{5}$. This problem required us to simplify square roots and combine like terms. By understanding the underlying concepts and applying the necessary mathematical operations, we were able to arrive at the correct solution.

Frequently Asked Questions

• Q: What is the square root of 45? A: The square root of 45 can be simplified as $3\sqrt{5}$.
• Q: How do I simplify the expression $11 \sqrt{45} - 4 \sqrt{5}$? A: To simplify the expression $11 \sqrt{45} - 4 \sqrt{5}$, you can substitute $\sqrt{45}$ with $3\sqrt{5}$, giving you $11 \times 3\sqrt{5} - 4 \sqrt{5}$. This simplifies to $33\sqrt{5} - 4\sqrt{5}$.
• Q: How do I combine like terms in the expression $33\sqrt{5} - 4\sqrt{5}$? A: To combine like terms in the expression $33\sqrt{5} - 4\sqrt{5}$, you can add the coefficients of the like terms. This gives you $(33 - 4)\sqrt{5}$, which simplifies to $29\sqrt{5}$.
• Q: What is the significance of this problem? A: This problem is significant because it requires us to apply mathematical operations, such as simplifying square roots and combining like terms. By solving this problem, we can develop our problem-solving skills and improve our understanding of mathematical concepts.
• Q: What are some real-world applications of this problem? A: This problem has real-world applications in various fields, such as engineering, physics, and computer science. For instance, in engineering, we may need to calculate the square root of a number to determine the length of a side of a triangle. In physics, we may need to calculate the square root of a number to determine the velocity of an object. In computer science, we may need to calculate the square root of a number to determine the distance between two points.
• Q: What are some common mistakes to avoid when solving this problem? A: Some common mistakes to avoid when solving this problem include:
  • Not simplifying the square root of 45
  • Not combining like terms
  • Not checking the units of the answer
• Q: What are some tips for solving this problem? A: Some tips for solving this problem include:
  • Simplifying the square root of 45
  • Combining like terms
  • Checking the units of the answer
• Q: What are some related problems to this problem? A: Some related problems to this problem include:
  • Simplifying expressions with square roots
  • Combining like terms
  • Checking the units of the answer
• Q: What are some resources for learning more about this problem? A: Some resources for learning more about this problem include:
  • Online math tutorials
  • Math textbooks
  • Online math communities