What Is The Simplified Form Of $\sqrt{64 X^{16}}$?A. $8 X^4$B. $ 8 X 8 8 X^8 8 X 8 [/tex]C. $32 X^4$D. $32 X^8$

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Understanding the Problem

When dealing with square roots, it's essential to remember that the square root of a number is a value that, when multiplied by itself, gives the original number. In mathematical terms, if $y = \sqrt{x}$, then $y^2 = x$. This concept is crucial in simplifying expressions involving square roots.

Breaking Down the Expression

The given expression is $\sqrt{64 x^{16}}$. To simplify this, we need to break it down into its prime factors. The number 64 can be expressed as $2^6$, and $x^{16}$ can be written as $(x4)4$.

Simplifying the Expression

Using the properties of exponents, we can rewrite the expression as $\sqrt{(2^6) (x4)4}$. Now, we can simplify the expression by taking the square root of the factors inside the square root.

Applying the Square Root Property

The square root of a product is equal to the product of the square roots. Therefore, we can rewrite the expression as $\sqrt{2^6} \cdot \sqrt{(x4)4}$. This simplifies to $2^3 \cdot (x4)2$.

Further Simplification

Using the property of exponents that states $(am)n = a^{m \cdot n}$, we can simplify $(x4)2$ to $x^8$. Therefore, the expression simplifies to $2^3 \cdot x^8$.

Evaluating the Final Answer

Now, we need to evaluate $2^3$, which is equal to 8. Therefore, the simplified form of $\sqrt{64 x^{16}}$ is $8 x^8$.

Conclusion

In conclusion, the simplified form of $\sqrt{64 x^{16}}$ is $8 x^8$. This is the correct answer among the given options.

Final Answer

The final answer is: 8x8\boxed{8 x^8}

Discussion

This problem requires a good understanding of the properties of exponents and square roots. The key concept used in this problem is the property of exponents that states $(am)n = a^{m \cdot n}$. This property is essential in simplifying expressions involving exponents.

Related Problems

If you're looking for more problems to practice, here are a few related problems:

  • Simplify the expression $\sqrt{81 x^8}$.
  • Simplify the expression $\sqrt{16 x^{12}}$.
  • Simplify the expression $\sqrt{25 x^{20}}$.

Tips and Tricks

Here are a few tips and tricks to help you solve problems like this:

  • Always start by breaking down the expression into its prime factors.
  • Use the properties of exponents to simplify the expression.
  • Remember that the square root of a product is equal to the product of the square roots.
  • Use the property of exponents that states $(am)n = a^{m \cdot n}$ to simplify expressions involving exponents.

Common Mistakes

Here are a few common mistakes to avoid when solving problems like this:

  • Not breaking down the expression into its prime factors.
  • Not using the properties of exponents to simplify the expression.
  • Not remembering that the square root of a product is equal to the product of the square roots.
  • Not using the property of exponents that states $(am)n = a^{m \cdot n}$ to simplify expressions involving exponents.

Conclusion

In conclusion, the simplified form of $\sqrt{64 x^{16}}$ is $8 x^8$. This problem requires a good understanding of the properties of exponents and square roots. By following the tips and tricks provided, you can avoid common mistakes and arrive at the correct answer.

Understanding the Basics

Simplifying expressions involving square roots can be a challenging task, but with the right approach, it can be made easier. In this article, we'll provide a Q&A section to help you understand the basics of simplifying expressions involving square roots.

Q: What is the simplified form of $\sqrt{64 x^{16}}$?

A: The simplified form of $\sqrt{64 x^{16}}$ is $8 x^8$. This is because $64 = 2^6$ and $x^{16} = (x4)4$, so we can rewrite the expression as $\sqrt{(2^6) (x4)4}$, which simplifies to $2^3 \cdot (x4)2$, and further simplifies to $8 x^8$.

Q: How do I simplify an expression involving a square root?

A: To simplify an expression involving a square root, you need to break it down into its prime factors. Then, use the properties of exponents to simplify the expression. Remember that the square root of a product is equal to the product of the square roots.

Q: What is the property of exponents that states $(am)n = a^{m \cdot n}$ used for?

A: This property is used to simplify expressions involving exponents. It states that when you raise a power to a power, you multiply the exponents. For example, $(x4)2 = x^{4 \cdot 2} = x^8$.

Q: How do I remember that the square root of a product is equal to the product of the square roots?

A: You can remember this by thinking of it as a "split" or "separate" the square root into two separate square roots. For example, $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$.

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

A: Some common mistakes to avoid include:

  • Not breaking down the expression into its prime factors.
  • Not using the properties of exponents to simplify the expression.
  • Not remembering that the square root of a product is equal to the product of the square roots.
  • Not using the property of exponents that states $(am)n = a^{m \cdot n}$ to simplify expressions involving exponents.

Q: How can I practice simplifying expressions involving square roots?

A: You can practice simplifying expressions involving square roots by working through problems like the ones provided in this article. You can also try creating your own problems and solving them.

Q: What are some tips for simplifying expressions involving square roots?

A: Some tips for simplifying expressions involving square roots include:

  • Always start by breaking down the expression into its prime factors.
  • Use the properties of exponents to simplify the expression.
  • Remember that the square root of a product is equal to the product of the square roots.
  • Use the property of exponents that states $(am)n = a^{m \cdot n}$ to simplify expressions involving exponents.

Q: How can I apply what I've learned to real-world problems?

A: You can apply what you've learned to real-world problems by using the concepts and techniques you've learned to simplify expressions involving square roots. For example, you might use these concepts to simplify expressions in physics, engineering, or other fields.

Q: What are some common applications of simplifying expressions involving square roots?

A: Some common applications of simplifying expressions involving square roots include:

  • Simplifying expressions in physics and engineering.
  • Simplifying expressions in algebra and calculus.
  • Simplifying expressions in computer science and programming.
  • Simplifying expressions in finance and economics.

Conclusion

In conclusion, simplifying expressions involving square roots can be a challenging task, but with the right approach, it can be made easier. By following the tips and tricks provided in this article, you can avoid common mistakes and arrive at the correct answer. Remember to always start by breaking down the expression into its prime factors, use the properties of exponents to simplify the expression, and remember that the square root of a product is equal to the product of the square roots.