Simplify The Expression: \[$\frac{-5}{9+\sqrt{6}}\$\]
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Introduction
Simplifying complex expressions is a crucial skill in mathematics, particularly in algebra and calculus. It involves breaking down intricate expressions into simpler forms, making them easier to understand and work with. In this article, we will focus on simplifying the expression {\frac{-5}{9+\sqrt{6}}$}$, which involves dealing with square roots and rational expressions.
Understanding the Expression
The given expression is {\frac{-5}{9+\sqrt{6}}$}$. To simplify this expression, we need to understand the concept of rational expressions and how to handle square roots in the denominator.
A rational expression is a fraction that contains variables or constants in the numerator and denominator. In this case, the numerator is a constant (-5), and the denominator is a variable expression (].
Simplifying the Expression
To simplify the expression, we need to rationalize the denominator, which involves getting rid of the square root in the denominator. We can do this by multiplying the numerator and denominator by the conjugate of the denominator.
The conjugate of (] is (]. By multiplying the numerator and denominator by this conjugate, we can eliminate the square root in the denominator.
Step-by-Step Solution
Step 1: Multiply the Numerator and Denominator by the Conjugate
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator:
{\frac{-5}{9+\sqrt{6}} \times \frac{9-\sqrt{6}}{9-\sqrt{6}}$}$
Step 2: Simplify the Expression
Now, we simplify the expression by multiplying the numerators and denominators:
{\frac{-5(9-\sqrt{6})}{(9+\sqrt{6})(9-\sqrt{6})}$}$
Step 3: Expand and Simplify
Next, we expand and simplify the expression:
{\frac{-45+5\sqrt{6}}{81-6}$}$
Step 4: Final Simplification
Finally, we simplify the expression by dividing the numerator and denominator by their greatest common divisor:
{\frac{-45+5\sqrt{6}}{75}$}$
Conclusion
Simplifying complex expressions like {\frac{-5}{9+\sqrt{6}}$}$ requires a step-by-step approach, involving rationalizing the denominator and simplifying the expression. By following these steps, we can break down intricate expressions into simpler forms, making them easier to understand and work with.
Tips and Tricks
- When dealing with square roots in the denominator, multiply the numerator and denominator by the conjugate of the denominator.
- Simplify the expression by expanding and combining like terms.
- Check for any common factors in the numerator and denominator and simplify accordingly.
Real-World Applications
Simplifying complex expressions has numerous real-world applications in fields such as engineering, physics, and computer science. It is essential to understand how to simplify expressions to solve problems in these fields.
Final Thoughts
Simplifying complex expressions is a crucial skill in mathematics, and it requires practice and patience to master. By following the steps outlined in this article, you can simplify expressions like {\frac{-5}{9+\sqrt{6}}$}$ and become more confident in your ability to tackle complex mathematical problems.
Additional Resources
- Khan Academy: Simplifying Rational Expressions
- Mathway: Simplifying Complex Expressions
- Wolfram Alpha: Simplifying Rational Expressions
Related Articles
- Simplifying Rational Expressions: A Step-by-Step Guide
- Rationalizing the Denominator: A Guide
- Simplifying Complex Fractions: A Guide
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Introduction
Simplifying complex expressions is a crucial skill in mathematics, particularly in algebra and calculus. In our previous article, we provided a step-by-step guide on how to simplify the expression {\frac{-5}{9+\sqrt{6}}$}$. In this article, we will answer some frequently asked questions (FAQs) related to simplifying complex expressions.
Q&A
Q: What is the purpose of rationalizing the denominator?
A: The purpose of rationalizing the denominator is to eliminate the square root in the denominator, making it easier to simplify the expression.
Q: How do I rationalize the denominator?
A: To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a binomial expression (] is (].
Q: What is the conjugate of a binomial expression?
A: The conjugate of a binomial expression (] is (]. For example, the conjugate of (] is (].
Q: How do I simplify a complex expression?
A: To simplify a complex expression, follow these steps:
- Multiply the numerator and denominator by the conjugate of the denominator.
- Simplify the expression by expanding and combining like terms.
- Check for any common factors in the numerator and denominator and simplify accordingly.
Q: What are some common mistakes to avoid when simplifying complex expressions?
A: Some common mistakes to avoid when simplifying complex expressions include:
- Not rationalizing the denominator
- Not simplifying the expression by expanding and combining like terms
- Not checking for any common factors in the numerator and denominator
Q: How do I know if an expression is already simplified?
A: An expression is already simplified if it cannot be simplified further by expanding and combining like terms.
Q: What are some real-world applications of simplifying complex expressions?
A: Simplifying complex expressions has numerous real-world applications in fields such as engineering, physics, and computer science. It is essential to understand how to simplify expressions to solve problems in these fields.
Tips and Tricks
- When dealing with square roots in the denominator, multiply the numerator and denominator by the conjugate of the denominator.
- Simplify the expression by expanding and combining like terms.
- Check for any common factors in the numerator and denominator and simplify accordingly.
Additional Resources
- Khan Academy: Simplifying Rational Expressions
- Mathway: Simplifying Complex Expressions
- Wolfram Alpha: Simplifying Rational Expressions
Related Articles
- Simplifying Rational Expressions: A Step-by-Step Guide
- Rationalizing the Denominator: A Guide
- Simplifying Complex Fractions: A Guide
Conclusion
Simplifying complex expressions is a crucial skill in mathematics, and it requires practice and patience to master. By following the steps outlined in this article and avoiding common mistakes, you can become more confident in your ability to tackle complex mathematical problems.
Final Thoughts
Simplifying complex expressions is a fundamental concept in mathematics, and it has numerous real-world applications. By understanding how to simplify expressions, you can solve problems in fields such as engineering, physics, and computer science.
Additional Tips
- Practice simplifying complex expressions regularly to become more confident in your ability to tackle complex mathematical problems.
- Use online resources such as Khan Academy, Mathway, and Wolfram Alpha to help you simplify complex expressions.
- Join a study group or find a study partner to help you practice simplifying complex expressions.