What Is The Difference When Simplified? 11 45 − 4 5 11 \sqrt{45} - 4 \sqrt{5} 11 45 ​ − 4 5 ​

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Understanding the Problem

When dealing with mathematical expressions involving square roots, it's essential to simplify them to make calculations easier and more manageable. The given expression $11 \sqrt{45} - 4 \sqrt{5}$ requires us to simplify the square roots and then perform the subtraction. To simplify the square roots, we need to factorize the numbers inside the square roots and look for perfect squares.

Simplifying Square Roots

To simplify $\sqrt{45}$, we can factorize 45 as $9 \times 5$. Since 9 is a perfect square, we can rewrite $\sqrt{45}$ as $\sqrt{9 \times 5}$, which simplifies to $3\sqrt{5}$.

Simplifying the Expression

Now that we have simplified $\sqrt{45}$ to $3\sqrt{5}$, we can substitute this value back into the original expression: $11 \sqrt{45} - 4 \sqrt{5}$. This becomes $11(3\sqrt{5}) - 4 \sqrt{5}$.

Distributing the Coefficient

To simplify the expression further, we need to distribute the coefficient 11 to the terms inside the parentheses: $11(3\sqrt{5})$. This gives us $33\sqrt{5}$.

Combining Like Terms

Now that we have simplified the expression to $33\sqrt{5} - 4 \sqrt{5}$, we can combine like terms. Since both terms have the same square root, we can subtract the coefficients: $33\sqrt{5} - 4 \sqrt{5} = (33 - 4)\sqrt{5}$.

Final Simplification

The final simplification of the expression is $(33 - 4)\sqrt{5} = 29\sqrt{5}$.

Conclusion

In conclusion, the difference when simplified is $29\sqrt{5}$. This is the final answer to the given expression $11 \sqrt{45} - 4 \sqrt{5}$.

Step-by-Step Solution

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

1. Simplify $\sqrt{45}$ to $3\sqrt{5}$.
2. Substitute the simplified value back into the original expression: $11 \sqrt{45} - 4 \sqrt{5}$.
3. Distribute the coefficient 11 to the terms inside the parentheses: $11(3\sqrt{5})$.
4. Simplify the expression to $33\sqrt{5} - 4 \sqrt{5}$.
5. Combine like terms by subtracting the coefficients: $(33 - 4)\sqrt{5}$.
6. Final simplification: $29\sqrt{5}$.

Frequently Asked Questions

• What is the difference when simplified?
• How do you simplify square roots?
• What is the final answer to the expression $11 \sqrt{45} - 4 \sqrt{5}$?

Related Topics

• Simplifying square roots
• Combining like terms
• Distributing coefficients

Final Answer

The final answer to the expression $11 \sqrt{45} - 4 \sqrt{5}$ is $\boxed{29\sqrt{5}}$.

Understanding Square Roots

Before we dive into the FAQs, 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.

Q&A

Q: What is the difference when simplified? $11 \sqrt{45} - 4 \sqrt{5}$

A: The difference when simplified is $29\sqrt{5}$.

Q: How do you simplify square roots?

A: To simplify square roots, you need to factorize the numbers inside the square roots and look for perfect squares. For example, $\sqrt{45}$ can be simplified to $3\sqrt{5}$ because 45 can be factorized as $9 \times 5$.

Q: What is the final answer to the expression $11 \sqrt{45} - 4 \sqrt{5}$?

A: The final answer to the expression $11 \sqrt{45} - 4 \sqrt{5}$ is $\boxed{29\sqrt{5}}$.

Q: How do you combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the terms with the same variable. For example, $33\sqrt{5} - 4 \sqrt{5}$ can be combined to $(33 - 4)\sqrt{5}$.

Q: What is the importance of simplifying square roots?

A: Simplifying square roots is important because it makes calculations easier and more manageable. It also helps to avoid errors and makes the solution more elegant.

Q: Can you provide a step-by-step solution to the problem?

A: Yes, here's a step-by-step solution to the problem:

1. Simplify $\sqrt{45}$ to $3\sqrt{5}$.
2. Substitute the simplified value back into the original expression: $11 \sqrt{45} - 4 \sqrt{5}$.
3. Distribute the coefficient 11 to the terms inside the parentheses: $11(3\sqrt{5})$.
4. Simplify the expression to $33\sqrt{5} - 4 \sqrt{5}$.
5. Combine like terms by subtracting the coefficients: $(33 - 4)\sqrt{5}$.
6. Final simplification: $29\sqrt{5}$.

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

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

• Not factorizing the numbers inside the square roots
• Not looking for perfect squares
• Not distributing the coefficients correctly
• Not combining like terms correctly

Q: Can you provide some examples of simplifying square roots?

A: Yes, here are some examples of simplifying square roots:

• $\sqrt{16}$ can be simplified to $4$ because 16 can be factorized as $4 \times 4$.
• $\sqrt{25}$ can be simplified to $5$ because 25 can be factorized as $5 \times 5$.
• $\sqrt{36}$ can be simplified to $6$ because 36 can be factorized as $6 \times 6$.

Conclusion

In conclusion, simplifying square roots is an essential skill in mathematics that can make calculations easier and more manageable. By following the steps outlined in this article, you can simplify square roots and solve problems with confidence.

Related Topics

• Simplifying square roots
• Combining like terms
• Distributing coefficients

Final Answer

The final answer to the expression $11 \sqrt{45} - 4 \sqrt{5}$ is $\boxed{29\sqrt{5}}$.