Simplify The Expression: 48 U 10 \sqrt{48 U^{10}} 48 U 10 ​ Assume That The Variable U U U Represents A Positive Number.

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Understanding the Problem

When simplifying the expression $\sqrt{48 u^{10}}$, we need to consider the properties of square roots and exponents. The expression involves a square root of a product of two terms: $48$ and $u^{10}$. Our goal is to simplify this expression by using the properties of square roots and exponents.

Breaking Down the Expression

To simplify the expression, we can start by breaking it down into its prime factors. The number $48$ can be factored as $2^4 \cdot 3$, and the variable $u^{10}$ can be written as $(u^2)^5$. Therefore, we can rewrite the expression as:

$$\sqrt{48 u^{10}} = \sqrt{2^4 \cdot 3 \cdot (u^2)^5}$$

Using the Properties of Square Roots

Now that we have broken down the expression into its prime factors, we can use the properties of square roots to simplify it further. The square root of a product can be written as the product of the square roots of the individual terms. Therefore, we can rewrite the expression as:

$$\sqrt{2^4 \cdot 3 \cdot (u^2)^5} = \sqrt{2^4} \cdot \sqrt{3} \cdot \sqrt{(u^2)^5}$$

Simplifying the Square Roots

Now that we have the expression in the form of a product of square roots, we can simplify each term individually. The square root of $2^4$ is $2^2$, the square root of $3$ is $\sqrt{3}$, and the square root of $(u^2)^5$ is $u^5$. Therefore, we can rewrite the expression as:

$$\sqrt{2^4} \cdot \sqrt{3} \cdot \sqrt{(u^2)^5} = 2^2 \cdot \sqrt{3} \cdot u^5$$

Final Simplification

Now that we have simplified each term individually, we can combine them to get the final simplified expression. Therefore, the final simplified expression is:

$$2^2 \cdot \sqrt{3} \cdot u^5 = 4 \cdot \sqrt{3} \cdot u^5$$

Conclusion

In this article, we have simplified the expression $\sqrt{48 u^{10}}$ by using the properties of square roots and exponents. We broke down the expression into its prime factors, used the properties of square roots to simplify it further, and finally combined the simplified terms to get the final expression. The final simplified expression is $4 \cdot \sqrt{3} \cdot u^5$.

Additional Tips and Tricks

When simplifying expressions involving square roots and exponents, it is essential to remember the following tips and tricks:

• Break down the expression into its prime factors to simplify it further.
• Use the properties of square roots to simplify the expression.
• Simplify each term individually before combining them.
• Be careful when simplifying expressions involving exponents, as the rules of exponents can be complex.

Common Mistakes to Avoid

When simplifying expressions involving square roots and exponents, it is essential to avoid the following common mistakes:

• Not breaking down the expression into its prime factors.
• Not using the properties of square roots to simplify the expression.
• Not simplifying each term individually before combining them.
• Not being careful when simplifying expressions involving exponents.

Real-World Applications

Simplifying expressions involving square roots and exponents has numerous real-world applications in various fields, including:

• Algebra: Simplifying expressions involving square roots and exponents is a fundamental concept in algebra.
• Calculus: Simplifying expressions involving square roots and exponents is essential in calculus, particularly in the study of limits and derivatives.
• Physics: Simplifying expressions involving square roots and exponents is crucial in physics, particularly in the study of motion and energy.
• Engineering: Simplifying expressions involving square roots and exponents is essential in engineering, particularly in the design and analysis of systems.

Final Thoughts

In conclusion, simplifying expressions involving square roots and exponents is a fundamental concept in mathematics that has numerous real-world applications. By following the tips and tricks outlined in this article, you can simplify expressions involving square roots and exponents with ease. Remember to break down the expression into its prime factors, use the properties of square roots to simplify it further, and simplify each term individually before combining them. With practice and patience, you can become proficient in simplifying expressions involving square roots and exponents.

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Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about simplifying the expression $\sqrt{48 u^{10}}$.

Q: What is the first step in simplifying the expression $\sqrt{48 u^{10}}$?

A: The first step in simplifying the expression $\sqrt{48 u^{10}}$ is to break it down into its prime factors. This involves factoring the number $48$ and the variable $u^{10}$.

Q: How do I factor the number $48$?

A: The number $48$ can be factored as $2^4 \cdot 3$. This means that $48$ can be expressed as the product of $2^4$ and $3$.

Q: How do I factor the variable $u^{10}$?

A: The variable $u^{10}$ can be factored as $(u^2)^5$. This means that $u^{10}$ can be expressed as the product of $(u^2)^5$.

Q: What is the next step in simplifying the expression $\sqrt{48 u^{10}}$?

A: The next step in simplifying the expression $\sqrt{48 u^{10}}$ is to use the properties of square roots to simplify it further. This involves rewriting the expression as the product of the square roots of the individual terms.

Q: How do I use the properties of square roots to simplify the expression $\sqrt{48 u^{10}}$?

A: To use the properties of square roots to simplify the expression $\sqrt{48 u^{10}}$, we can rewrite it as:

$$\sqrt{48 u^{10}} = \sqrt{2^4 \cdot 3 \cdot (u^2)^5}$$

This allows us to simplify the expression further by taking the square root of each term individually.

Q: What is the final simplified expression?

A: The final simplified expression is $4 \cdot \sqrt{3} \cdot u^5$.

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

A: Some common mistakes to avoid when simplifying expressions involving square roots and exponents include:

• Not breaking down the expression into its prime factors.
• Not using the properties of square roots to simplify the expression.
• Not simplifying each term individually before combining them.
• Not being careful when simplifying expressions involving exponents.

Q: What are some real-world applications of simplifying expressions involving square roots and exponents?

A: Simplifying expressions involving square roots and exponents has numerous real-world applications in various fields, including:

• Algebra: Simplifying expressions involving square roots and exponents is a fundamental concept in algebra.
• Calculus: Simplifying expressions involving square roots and exponents is essential in calculus, particularly in the study of limits and derivatives.
• Physics: Simplifying expressions involving square roots and exponents is crucial in physics, particularly in the study of motion and energy.
• Engineering: Simplifying expressions involving square roots and exponents is essential in engineering, particularly in the design and analysis of systems.

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

A: You can practice simplifying expressions involving square roots and exponents by working through examples and exercises. You can also try simplifying expressions involving square roots and exponents on your own, using the tips and tricks outlined in this article.

Conclusion

In this article, we have answered some of the most frequently asked questions about simplifying the expression $\sqrt{48 u^{10}}$. We have covered topics such as breaking down the expression into its prime factors, using the properties of square roots to simplify it further, and avoiding common mistakes. We have also discussed some real-world applications of simplifying expressions involving square roots and exponents. By following the tips and tricks outlined in this article, you can simplify expressions involving square roots and exponents with ease.