What Is The Simplified Form Of 10 , 000 X 64 \sqrt{10,000 X^{64}} 10 , 000 X 64 ​ ?A. 5000 X 32 5000 X^{32} 5000 X 32 B. 5000 X 8 5000 X^8 5000 X 8 C. 100 X 8 100 X^8 100 X 8 D. 100 X 32 100 X^{32} 100 X 32

Understanding the Problem

The problem requires us to simplify the expression $\sqrt{10,000 x^{64}}$. To do this, we need to understand the properties of square roots and how they interact with exponents. The square root of a number is a value that, when multiplied by itself, gives the original number. In mathematical terms, $\sqrt{a} \times \sqrt{a} = a$. We will use this property to simplify the given expression.

Breaking Down the Expression

The expression $\sqrt{10,000 x^{64}}$ can be broken down into two parts: $\sqrt{10,000}$ and $\sqrt{x^{64}}$. We can simplify each part separately and then combine them to get the final result.

Simplifying the First Part

The first part is $\sqrt{10,000}$. To simplify this, we need to find the largest perfect square that divides 10,000. A perfect square is a number that can be expressed as the square of an integer. In this case, the largest perfect square that divides 10,000 is 100, which is equal to $10^2$. Therefore, $\sqrt{10,000} = \sqrt{10^2 \times 100} = 10\sqrt{100} = 10 \times 10 = 100$.

Simplifying the Second Part

The second part is $\sqrt{x^{64}}$. To simplify this, we need to use the property of exponents that states $\sqrt{x^n} = x^{n/2}$. In this case, $n = 64$, so $\sqrt{x^{64}} = x^{64/2} = x^{32}$.

Combining the Parts

Now that we have simplified both parts, we can combine them to get the final result. $\sqrt{10,000 x^{64}} = 100 \times x^{32} = 100x^{32}$.

Conclusion

The simplified form of $\sqrt{10,000 x^{64}}$ is $100x^{32}$. This is the correct answer among the options provided.

Comparison with Options

Let's compare our result with the options provided:

• A. $5000 x^{32}$: This is not the correct answer because the coefficient is 5000, not 100.
• B. $5000 x^8$: This is not the correct answer because the exponent is 8, not 32.
• C. $100 x^8$: This is not the correct answer because the exponent is 8, not 32.
• D. $100 x^{32}$: This is the correct answer.

Final Answer

The final answer is $\boxed{100x^{32}}$.

Understanding Square Roots and Exponents

Q: What is the square root of a number?

A: The square root of a number is a value that, when multiplied by itself, gives the original number. In mathematical terms, $\sqrt{a} \times \sqrt{a} = a$.

Q: What is the property of exponents that states $\sqrt{x^n} = x^{n/2}$?

A: This property states that the square root of a number raised to a power is equal to the number raised to half of that power. For example, $\sqrt{x^4} = x^{4/2} = x^2$.

Simplifying Square Roots with Exponents

Q: How do you simplify $\sqrt{10,000 x^{64}}$?

A: To simplify this expression, we need to break it down into two parts: $\sqrt{10,000}$ and $\sqrt{x^{64}}$. We can simplify each part separately and then combine them to get the final result.

Q: How do you simplify $\sqrt{10,000}$?

A: The largest perfect square that divides 10,000 is 100, which is equal to $10^2$. Therefore, $\sqrt{10,000} = \sqrt{10^2 \times 100} = 10\sqrt{100} = 10 \times 10 = 100$.

Q: How do you simplify $\sqrt{x^{64}}$?

A: We can use the property of exponents that states $\sqrt{x^n} = x^{n/2}$. In this case, $n = 64$, so $\sqrt{x^{64}} = x^{64/2} = x^{32}$.

Q: What is the final result of simplifying $\sqrt{10,000 x^{64}}$?

A: The final result is $100x^{32}$.

Common Mistakes to Avoid

Q: What is a common mistake when simplifying square roots with exponents?

A: A common mistake is to forget to simplify the square root of the number raised to a power. For example, $\sqrt{x^4}$ is not equal to $x^2$, but rather $x^2$ is the correct result after simplifying the square root.

Q: How can you avoid this mistake?

A: To avoid this mistake, make sure to use the property of exponents that states $\sqrt{x^n} = x^{n/2}$ and simplify the square root of the number raised to a power.

Conclusion

Simplifying square roots with exponents can be a challenging task, but by understanding the properties of square roots and exponents, you can simplify expressions like $\sqrt{10,000 x^{64}}$ to get the final result of $100x^{32}$. Remember to avoid common mistakes like forgetting to simplify the square root of the number raised to a power.

Final Tips

• Always use the property of exponents that states $\sqrt{x^n} = x^{n/2}$ when simplifying square roots with exponents.
• Make sure to simplify the square root of the number raised to a power.
• Use the largest perfect square that divides the number to simplify the square root.

Final Answer

The final answer is $\boxed{100x^{32}}$.