Simplify The Expression: $\[ -11 \sqrt{1215} \\]

$$

Understanding the Problem

When dealing with square roots, it's essential to simplify the expression by finding the largest perfect square that divides the number inside the square root. In this case, we have the expression $-11 \sqrt{1215}$, and our goal is to simplify it.

Breaking Down the Number Inside the Square Root

To simplify the expression, we need to break down the number inside the square root, which is 1215. We can start by finding the prime factorization of 1215.

Prime Factorization of 1215

The prime factorization of 1215 is:

1215 = 3^4 × 5

This means that 1215 can be expressed as the product of 3 raised to the power of 4 and 5.

Simplifying the Square Root

Now that we have the prime factorization of 1215, we can simplify the square root by taking out the largest perfect square that divides 1215.

Simplifying the Square Root of 1215

The largest perfect square that divides 1215 is 3^2, which is equal to 9. Therefore, we can simplify the square root of 1215 as follows:

√1215 = √(3^4 × 5) = √(3^2 × 3^2 × 5) = 3^2 × √(3^2 × 5) = 9 × √(3^2 × 5) = 9 × √(9 × 5) = 9 × √(45)

Simplifying the Square Root of 45

Now that we have simplified the square root of 1215, we can further simplify the square root of 45.

The prime factorization of 45 is:

45 = 3^2 × 5

Therefore, we can simplify the square root of 45 as follows:

√45 = √(3^2 × 5) = 3 × √5

Simplifying the Original Expression

Now that we have simplified the square root of 1215, we can simplify the original expression $-11 \sqrt{1215}$.

Simplifying the Original Expression

We can simplify the original expression as follows:

$-11 \sqrt{1215}$ = $-11 \sqrt{(3^4 × 5)}$ = $-11 (3^2 × √(3^2 × 5))$ = $-11 (9 × √(9 × 5))$ = $-11 (9 × √(45))$ = $-11 (9 × 3 × √5)$ = $-11 (27 × √5)$ = $-297 \sqrt{5}$

Conclusion

In conclusion, we have simplified the expression $-11 \sqrt{1215}$ by breaking down the number inside the square root, finding the largest perfect square that divides it, and simplifying the square root. The simplified expression is $-297 \sqrt{5}$.

Final Answer

The final answer is: $\boxed{-297 \sqrt{5}}$

$$

Understanding the Problem

When dealing with square roots, it's essential to simplify the expression by finding the largest perfect square that divides the number inside the square root. In this case, we have the expression $-11 \sqrt{1215}$, and our goal is to simplify it.

Q&A

Q: What is the prime factorization of 1215?

A: The prime factorization of 1215 is 3^4 × 5.

Q: How do we simplify the square root of 1215?

A: We simplify the square root of 1215 by taking out the largest perfect square that divides 1215. In this case, the largest perfect square that divides 1215 is 3^2, which is equal to 9.

Q: How do we simplify the square root of 45?

A: We simplify the square root of 45 by taking out the largest perfect square that divides 45. In this case, the largest perfect square that divides 45 is 3^2, which is equal to 9.

Q: What is the simplified expression for $-11 \sqrt{1215}$?

A: The simplified expression for $-11 \sqrt{1215}$ is $-297 \sqrt{5}$.

Q: Why is it essential to simplify the expression by finding the largest perfect square that divides the number inside the square root?

A: It's essential to simplify the expression by finding the largest perfect square that divides the number inside the square root because it helps us to reduce the complexity of the expression and make it easier to work with.

Q: What is the final answer to the problem?

A: The final answer to the problem is $-297 \sqrt{5}$.

Common Mistakes to Avoid

When simplifying the expression $-11 \sqrt{1215}$, there are several common mistakes to avoid:

• Not breaking down the number inside the square root into its prime factors.
• Not finding the largest perfect square that divides the number inside the square root.
• Not simplifying the square root of the number inside the square root.

Tips for Simplifying Square Roots

When simplifying square roots, here are some tips to keep in mind:

• Break down the number inside the square root into its prime factors.
• Find the largest perfect square that divides the number inside the square root.
• Simplify the square root of the number inside the square root.
• Use the properties of square roots to simplify the expression.

Conclusion

In conclusion, simplifying the expression $-11 \sqrt{1215}$ requires breaking down the number inside the square root into its prime factors, finding the largest perfect square that divides the number inside the square root, and simplifying the square root. By following these steps and avoiding common mistakes, we can simplify the expression and arrive at the final answer of $-297 \sqrt{5}$.

Final Answer

The final answer is: $\boxed{-297 \sqrt{5}}$