Simplify The Expression:$\[ \frac{\sqrt{98} - \sqrt{50}}{\sqrt{32}} \\]

Introduction

Radical expressions can be complex and intimidating, but with the right approach, they can be simplified to reveal their underlying structure. In this article, we will focus on simplifying the expression $\frac{\sqrt{98} - \sqrt{50}}{\sqrt{32}}$. We will break down the problem into manageable steps, using techniques such as factoring and simplifying radicals.

Understanding the Expression

The given expression involves three square roots: $\sqrt{98}$, $\sqrt{50}$, and $\sqrt{32}$. To simplify the expression, we need to understand the properties of square roots and how they can be manipulated.

Properties of Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because $4 \times 4 = 16$. The square root of a number can be positive or negative, but in this case, we will focus on the positive square root.

Simplifying the Expression

To simplify the expression, we can start by factoring the numbers inside the square roots.

Factoring the Numbers

• $\sqrt{98}$ can be factored as $\sqrt{2 \times 49}$, which simplifies to $7\sqrt{2}$.
• $\sqrt{50}$ can be factored as $\sqrt{2 \times 25}$, which simplifies to $5\sqrt{2}$.
• $\sqrt{32}$ can be factored as $\sqrt{2 \times 16}$, which simplifies to $4\sqrt{2}$.

Simplifying the Expression

Now that we have factored the numbers inside the square roots, we can substitute the simplified expressions back into the original expression.

$\frac{\sqrt{98} - \sqrt{50}}{\sqrt{32}} = \frac{7\sqrt{2} - 5\sqrt{2}}{4\sqrt{2}}$

Combining Like Terms

We can combine the like terms in the numerator.

$\frac{7\sqrt{2} - 5\sqrt{2}}{4\sqrt{2}} = \frac{2\sqrt{2}}{4\sqrt{2}}$

Canceling Out Common Factors

We can cancel out the common factor of $\sqrt{2}$ in the numerator and denominator.

$\frac{2\sqrt{2}}{4\sqrt{2}} = \frac{2}{4}$

Simplifying the Fraction

We can simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 2.

$\frac{2}{4} = \frac{1}{2}$

Conclusion

Simplifying radical expressions requires a step-by-step approach, using techniques such as factoring and simplifying radicals. By breaking down the problem into manageable steps, we can reveal the underlying structure of the expression and simplify it to its simplest form. In this article, we simplified the expression $\frac{\sqrt{98} - \sqrt{50}}{\sqrt{32}}$ to $\frac{1}{2}$.

Additional Tips and Tricks

• When simplifying radical expressions, it's essential to factor the numbers inside the square roots to reveal their underlying structure.
• Use the properties of square roots to simplify the expression, such as combining like terms and canceling out common factors.
• Practice simplifying radical expressions to become more comfortable with the techniques and to develop your problem-solving skills.

Common Mistakes to Avoid

• Don't forget to factor the numbers inside the square roots to reveal their underlying structure.
• Be careful when combining like terms and canceling out common factors to avoid making mistakes.
• Make sure to simplify the fraction by dividing both the numerator and denominator by their greatest common divisor.

Real-World Applications

Simplifying radical expressions has numerous real-world applications, such as:

• Engineering: Simplifying radical expressions is essential in engineering, where complex mathematical expressions are used to design and analyze systems.
• Physics: Simplifying radical expressions is crucial in physics, where complex mathematical expressions are used to describe the behavior of physical systems.
• Computer Science: Simplifying radical expressions is essential in computer science, where complex mathematical expressions are used to develop algorithms and data structures.

Conclusion

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Introduction

Simplifying radical expressions can be a challenging task, but with the right approach, it can be made easier. In this article, we will provide a Q&A guide to help you understand the concepts and techniques involved in simplifying radical expressions.

Q: What is a radical expression?

A: A radical expression is an expression that contains a square root or a higher root of a number. For example, $\sqrt{16}$ is a radical expression.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to follow these steps:

1. Factor the numbers inside the square root to reveal their underlying structure.
2. Use the properties of square roots to simplify the expression, such as combining like terms and canceling out common factors.
3. Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor.

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

A: Some common mistakes to avoid when simplifying radical expressions include:

• Forgetting to factor the numbers inside the square root.
• Not combining like terms correctly.
• Not canceling out common factors correctly.
• Not simplifying the fraction correctly.

Q: How do I factor numbers inside a square root?

A: To factor numbers inside a square root, you need to find the prime factors of the number. For example, to factor 98, you can write it as $2 \times 49$, which simplifies to $2 \times 7^2$.

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

A: Simplifying radical expressions has numerous real-world applications, including:

• Engineering: Simplifying radical expressions is essential in engineering, where complex mathematical expressions are used to design and analyze systems.
• Physics: Simplifying radical expressions is crucial in physics, where complex mathematical expressions are used to describe the behavior of physical systems.
• Computer Science: Simplifying radical expressions is essential in computer science, where complex mathematical expressions are used to develop algorithms and data structures.

Q: How do I simplify a radical expression with multiple square roots?

A: To simplify a radical expression with multiple square roots, you need to follow these steps:

1. Factor the numbers inside each square root to reveal their underlying structure.
2. Use the properties of square roots to simplify the expression, such as combining like terms and canceling out common factors.
3. Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor.

Q: What are some tips and tricks for simplifying radical expressions?

A: Some tips and tricks for simplifying radical expressions include:

• Use the properties of square roots to simplify the expression, such as combining like terms and canceling out common factors.
• Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor.
• Practice simplifying radical expressions to become more comfortable with the techniques and to develop your problem-solving skills.

Conclusion

Simplifying radical expressions is a crucial skill in mathematics, with numerous real-world applications. By following the steps outlined in this Q&A guide, you can simplify radical expressions and develop your problem-solving skills. Remember to factor numbers inside the square root, use the properties of square roots to simplify the expression, and simplify the fraction by dividing both the numerator and denominator by their greatest common divisor.

Additional Resources

• Math textbooks: Check out math textbooks for more information on simplifying radical expressions.
• Online resources: Visit online resources, such as Khan Academy and Mathway, for more information on simplifying radical expressions.
• Practice problems: Practice simplifying radical expressions with practice problems to develop your problem-solving skills.

Common Mistakes to Avoid

• Forgetting to factor the numbers inside the square root.
• Not combining like terms correctly.
• Not canceling out common factors correctly.
• Not simplifying the fraction correctly.

Real-World Applications

• Engineering: Simplifying radical expressions is essential in engineering, where complex mathematical expressions are used to design and analyze systems.
• Physics: Simplifying radical expressions is crucial in physics, where complex mathematical expressions are used to describe the behavior of physical systems.
• Computer Science: Simplifying radical expressions is essential in computer science, where complex mathematical expressions are used to develop algorithms and data structures.