Simplify The Expression: 169 X 6 \sqrt{169 X^6} 169 X 6 ​

$$

Understanding the Problem

When dealing with square roots, it's essential to simplify the expression by factoring out perfect squares. In this case, we have the expression $\sqrt{169 x^6}$, and our goal is to simplify it.

Breaking Down the Expression

The expression $\sqrt{169 x^6}$ can be broken down into two parts: the number under the square root, which is 169, and the variable part, which is $x^6$. We can start by simplifying the number under the square root.

Simplifying the Number Under the Square Root

The number 169 is a perfect square because it can be expressed as the square of an integer. Specifically, $13^2 = 169$. Therefore, we can rewrite the expression as $\sqrt{(13^2) x^6}$.

Applying the Property of Square Roots

One of the properties of square roots is that we can take the square root of a product by taking the square root of each factor. In this case, we can rewrite the expression as $\sqrt{13^2} \cdot \sqrt{x^6}$.

Simplifying the Square Root of a Perfect Square

The square root of a perfect square is equal to the number itself. Therefore, $\sqrt{13^2} = 13$. So, we can simplify the expression to $13 \cdot \sqrt{x^6}$.

Simplifying the Variable Part

Now, let's focus on the variable part, which is $x^6$. We can rewrite it as $(x^3)^2$. This is because $x^6$ can be expressed as the square of $x^3$.

Applying the Property of Square Roots Again

Using the property of square roots that we mentioned earlier, we can take the square root of a product by taking the square root of each factor. In this case, we can rewrite the expression as $13 \cdot \sqrt{(x^3)^2}$.

Simplifying the Square Root of a Perfect Square Again

As we mentioned earlier, the square root of a perfect square is equal to the number itself. Therefore, $\sqrt{(x^3)^2} = x^3$. So, we can simplify the expression to $13x^3$.

Conclusion

In conclusion, we have simplified the expression $\sqrt{169 x^6}$ to $13x^3$. This is the final simplified form of the expression.

Final Answer

The final answer is: $\boxed{13x^3}$

Step-by-Step Solution

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

1. Break down the expression into two parts: the number under the square root and the variable part.
2. Simplify the number under the square root by finding its square root.
3. Apply the property of square roots to take the square root of each factor.
4. Simplify the square root of a perfect square.
5. Rewrite the variable part as a perfect square.
6. Apply the property of square roots again to take the square root of each factor.
7. Simplify the square root of a perfect square again.
8. Combine the simplified parts to get the final answer.

Common Mistakes to Avoid

Here are some common mistakes to avoid when simplifying expressions with square roots:

• Not recognizing perfect squares.
• Not applying the property of square roots correctly.
• Not simplifying the variable part correctly.
• Not combining the simplified parts correctly.

Real-World Applications

Simplifying expressions with square roots has many real-world applications, such as:

• Calculating distances and heights in geometry and trigonometry.
• Solving problems in physics and engineering.
• Working with algebraic expressions in computer science.

Tips and Tricks

Here are some tips and tricks to help you simplify expressions with square roots:

• Always look for perfect squares.
• Use the property of square roots to simplify expressions.
• Simplify the variable part correctly.
• Combine the simplified parts correctly.

Practice Problems

Here are some practice problems to help you practice simplifying expressions with square roots:

• Simplify the expression $\sqrt{25 x^4}$.
• Simplify the expression $\sqrt{144 y^8}$.
• Simplify the expression $\sqrt{49 z^6}$.

Conclusion

In conclusion, simplifying expressions with square roots is an essential skill in mathematics. By following the steps outlined in this article, you can simplify expressions with square roots and apply them to real-world problems. Remember to always look for perfect squares, use the property of square roots, simplify the variable part correctly, and combine the simplified parts correctly.

$$

Frequently Asked Questions

Q: What is the final simplified form of the expression $\sqrt{169 x^6}$?

A: The final simplified form of the expression $\sqrt{169 x^6}$ is $13x^3$.

Q: How do I simplify the expression $\sqrt{169 x^6}$?

A: To simplify the expression $\sqrt{169 x^6}$, you need to break it down into two parts: the number under the square root and the variable part. Then, simplify the number under the square root by finding its square root, apply the property of square roots to take the square root of each factor, simplify the square root of a perfect square, rewrite the variable part as a perfect square, apply the property of square roots again to take the square root of each factor, simplify the square root of a perfect square again, and combine the simplified parts to get the final answer.

Q: What is the property of square roots that I need to apply to simplify the expression $\sqrt{169 x^6}$?

A: The property of square roots that you need to apply is that you can take the square root of a product by taking the square root of each factor. This means that you can rewrite the expression as $\sqrt{13^2} \cdot \sqrt{x^6}$.

Q: How do I simplify the variable part $x^6$?

A: To simplify the variable part $x^6$, you need to rewrite it as a perfect square. In this case, you can rewrite it as $(x^3)^2$.

Q: What is the final simplified form of the variable part $x^6$?

A: The final simplified form of the variable part $x^6$ is $x^3$.

Q: How do I combine the simplified parts to get the final answer?

A: To combine the simplified parts, you need to multiply the simplified number part and the simplified variable part. In this case, you need to multiply $13$ and $x^3$ to get the final answer $13x^3$.

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

A: Some common mistakes to avoid when simplifying expressions with square roots include not recognizing perfect squares, not applying the property of square roots correctly, not simplifying the variable part correctly, and not combining the simplified parts correctly.

Q: What are some real-world applications of simplifying expressions with square roots?

A: Some real-world applications of simplifying expressions with square roots include calculating distances and heights in geometry and trigonometry, solving problems in physics and engineering, and working with algebraic expressions in computer science.

Q: What are some tips and tricks to help me simplify expressions with square roots?

A: Some tips and tricks to help you simplify expressions with square roots include always looking for perfect squares, using the property of square roots to simplify expressions, simplifying the variable part correctly, and combining the simplified parts correctly.

Q: What are some practice problems to help me practice simplifying expressions with square roots?

A: Some practice problems to help you practice simplifying expressions with square roots include simplifying the expression $\sqrt{25 x^4}$, simplifying the expression $\sqrt{144 y^8}$, and simplifying the expression $\sqrt{49 z^6}$.

Additional Resources

• For more information on simplifying expressions with square roots, check out the following resources:

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

Conclusion

In conclusion, simplifying expressions with square roots is an essential skill in mathematics. By following the steps outlined in this article and practicing with the practice problems, you can simplify expressions with square roots and apply them to real-world problems. Remember to always look for perfect squares, use the property of square roots, simplify the variable part correctly, and combine the simplified parts correctly.