Simplify The Expression:$\[ \frac{2-\sqrt{48}}{6} \\]

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of various mathematical concepts, including radicals, fractions, and algebraic manipulations. In this article, we will focus on simplifying the given expression $\frac{2-\sqrt{48}}{6}$, which involves radicals and fractions. We will break down the expression into manageable steps, and we will use various mathematical techniques to simplify it.

Understanding the Expression

The given expression is $\frac{2-\sqrt{48}}{6}$. To simplify this expression, we need to understand the properties of radicals and fractions. A radical is a mathematical operation that involves the extraction of the square root of a number. In this expression, we have a radical in the numerator, which is $\sqrt{48}$. We also have a fraction in the expression, which is $\frac{2-\sqrt{48}}{6}$.

Simplifying the Radical

To simplify the radical $\sqrt{48}$, we need to find the prime factorization of 48. The prime factorization of 48 is $2^4 \cdot 3$. We can rewrite the radical as $\sqrt{2^4 \cdot 3}$. Using the property of radicals that $\sqrt{a^2} = a$, we can simplify the radical as $2\sqrt{3}$.

Simplifying the Expression

Now that we have simplified the radical, we can simplify the expression. We can rewrite the expression as $\frac{2-2\sqrt{3}}{6}$. To simplify this expression, we can use the distributive property of fractions, which states that $\frac{a-b}{c} = \frac{a}{c} - \frac{b}{c}$. Using this property, we can rewrite the expression as $\frac{2}{6} - \frac{2\sqrt{3}}{6}$.

Further Simplification

We can further simplify the expression by combining the two fractions. We can rewrite the expression as $\frac{2-2\sqrt{3}}{6}$. To simplify this expression, we can use the property of fractions that $\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}$. Using this property, we can rewrite the expression as $\frac{2(1-\sqrt{3})}{6}$.

Final Simplification

We can further simplify the expression by canceling out the common factor of 2 in the numerator and denominator. We can rewrite the expression as $\frac{1-\sqrt{3}}{3}$.

Conclusion

In this article, we simplified the expression $\frac{2-\sqrt{48}}{6}$ using various mathematical techniques, including radicals and fractions. We broke down the expression into manageable steps and used various mathematical properties to simplify it. We simplified the radical $\sqrt{48}$ as $2\sqrt{3}$ and then simplified the expression as $\frac{1-\sqrt{3}}{3}$. This expression is the final simplified form of the given expression.

Tips and Tricks

• When simplifying radicals, it is essential to find the prime factorization of the number inside the radical.
• When simplifying fractions, it is essential to use the distributive property of fractions.
• When simplifying expressions, it is essential to combine like terms and cancel out common factors.

Real-World Applications

Simplifying algebraic expressions is a crucial skill in mathematics, and it has numerous real-world applications. For example, in physics, algebraic expressions are used to describe the motion of objects. In engineering, algebraic expressions are used to design and optimize systems. In economics, algebraic expressions are used to model and analyze economic systems.

Common Mistakes

• When simplifying radicals, it is easy to make mistakes by not finding the prime factorization of the number inside the radical.
• When simplifying fractions, it is easy to make mistakes by not using the distributive property of fractions.
• When simplifying expressions, it is easy to make mistakes by not combining like terms and canceling out common factors.

Final Thoughts

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of various mathematical concepts, including radicals, fractions, and algebraic manipulations. In this article, we simplified the expression $\frac{2-\sqrt{48}}{6}$ using various mathematical techniques, including radicals and fractions. We broke down the expression into manageable steps and used various mathematical properties to simplify it. We simplified the radical $\sqrt{48}$ as $2\sqrt{3}$ and then simplified the expression as $\frac{1-\sqrt{3}}{3}$. This expression is the final simplified form of the given expression.

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Introduction

In our previous article, we simplified the expression $\frac{2-\sqrt{48}}{6}$ using various mathematical techniques, including radicals and fractions. We broke down the expression into manageable steps and used various mathematical properties to simplify it. In this article, we will answer some of the most frequently asked questions about simplifying the expression $\frac{2-\sqrt{48}}{6}$.

Q&A

Q: What is the prime factorization of 48?

A: The prime factorization of 48 is $2^4 \cdot 3$.

Q: How do I simplify the radical $\sqrt{48}$?

A: To simplify the radical $\sqrt{48}$, you need to find the prime factorization of 48, which is $2^4 \cdot 3$. Then, you can rewrite the radical as $\sqrt{2^4 \cdot 3}$ and simplify it as $2\sqrt{3}$.

Q: How do I simplify the expression $\frac{2-\sqrt{48}}{6}$?

A: To simplify the expression $\frac{2-\sqrt{48}}{6}$, you need to simplify the radical $\sqrt{48}$ first. Then, you can rewrite the expression as $\frac{2-2\sqrt{3}}{6}$ and simplify it further by combining the two fractions.

Q: What is the final simplified form of the expression $\frac{2-\sqrt{48}}{6}$?

A: The final simplified form of the expression $\frac{2-\sqrt{48}}{6}$ is $\frac{1-\sqrt{3}}{3}$.

Q: What are some common mistakes to avoid when simplifying radicals?

A: Some common mistakes to avoid when simplifying radicals include not finding the prime factorization of the number inside the radical, not using the property of radicals that $\sqrt{a^2} = a$, and not simplifying the radical correctly.

Q: What are some common mistakes to avoid when simplifying fractions?

A: Some common mistakes to avoid when simplifying fractions include not using the distributive property of fractions, not combining like terms, and not canceling out common factors.

Q: What are some real-world applications of simplifying algebraic expressions?

A: Simplifying algebraic expressions has numerous real-world applications, including physics, engineering, and economics. In physics, algebraic expressions are used to describe the motion of objects. In engineering, algebraic expressions are used to design and optimize systems. In economics, algebraic expressions are used to model and analyze economic systems.

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by working on problems and exercises that involve radicals, fractions, and algebraic manipulations. You can also use online resources and tools to help you practice and improve your skills.

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of various mathematical concepts, including radicals, fractions, and algebraic manipulations. In this article, we answered some of the most frequently asked questions about simplifying the expression $\frac{2-\sqrt{48}}{6}$. We hope that this article has been helpful in providing you with a better understanding of how to simplify algebraic expressions.

Tips and Tricks

• When simplifying radicals, it is essential to find the prime factorization of the number inside the radical.
• When simplifying fractions, it is essential to use the distributive property of fractions.
• When simplifying expressions, it is essential to combine like terms and cancel out common factors.
• Practice is key to improving your skills in simplifying algebraic expressions.
• Use online resources and tools to help you practice and improve your skills.

Real-World Applications

Simplifying algebraic expressions has numerous real-world applications, including physics, engineering, and economics. In physics, algebraic expressions are used to describe the motion of objects. In engineering, algebraic expressions are used to design and optimize systems. In economics, algebraic expressions are used to model and analyze economic systems.

Common Mistakes

• When simplifying radicals, it is easy to make mistakes by not finding the prime factorization of the number inside the radical.
• When simplifying fractions, it is easy to make mistakes by not using the distributive property of fractions.
• When simplifying expressions, it is easy to make mistakes by not combining like terms and canceling out common factors.

Final Thoughts

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of various mathematical concepts, including radicals, fractions, and algebraic manipulations. In this article, we answered some of the most frequently asked questions about simplifying the expression $\frac{2-\sqrt{48}}{6}$. We hope that this article has been helpful in providing you with a better understanding of how to simplify algebraic expressions.