Simplify The Expression: 1 5 − 3 \frac{1}{5-\sqrt{3}} 5 − 3 ​ 1 ​

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Introduction

When dealing with expressions that involve square roots in the denominator, it's essential to rationalize the denominator to simplify the expression. Rationalizing the denominator involves getting rid of the square root in the denominator by multiplying the numerator and denominator by a cleverly chosen value. In this article, we'll explore how to rationalize the denominator of the expression 15−3\frac{1}{5-\sqrt{3}} and simplify it to its simplest form.

Understanding the Concept of Rationalizing the Denominator

Rationalizing the denominator is a technique used to simplify expressions that involve square roots in the denominator. The goal is to eliminate the square root in the denominator by multiplying the numerator and denominator by a value that will cancel out the square root. This technique is essential in algebra and is used to simplify expressions that involve square roots.

The Importance of Rationalizing the Denominator

Rationalizing the denominator is crucial in algebra because it allows us to simplify expressions that involve square roots in the denominator. When we rationalize the denominator, we can eliminate the square root in the denominator, making it easier to work with the expression. This technique is also used in calculus and other advanced math topics.

Step-by-Step Guide to Rationalizing the Denominator

To rationalize the denominator of the expression 15−3\frac{1}{5-\sqrt{3}}, we need to follow these steps:

Step 1: Identify the Value to Multiply

To rationalize the denominator, we need to multiply the numerator and denominator by a value that will cancel out the square root in the denominator. In this case, we need to multiply by the conjugate of the denominator, which is 5+35+\sqrt{3}.

Step 2: Multiply the Numerator and Denominator

Now that we have identified the value to multiply, we can multiply the numerator and denominator by 5+35+\sqrt{3}.

15−3⋅5+35+3\frac{1}{5-\sqrt{3}} \cdot \frac{5+\sqrt{3}}{5+\sqrt{3}}

Step 3: Simplify the Expression

Now that we have multiplied the numerator and denominator, we can simplify the expression.

15−3⋅5+35+3=5+3(5−3)(5+3)\frac{1}{5-\sqrt{3}} \cdot \frac{5+\sqrt{3}}{5+\sqrt{3}} = \frac{5+\sqrt{3}}{(5-\sqrt{3})(5+\sqrt{3})}

Step 4: Simplify the Denominator

To simplify the denominator, we can use the difference of squares formula, which states that (a−b)(a+b)=a2−b2(a-b)(a+b) = a^2 - b^2.

5+3(5−3)(5+3)=5+352−(3)2\frac{5+\sqrt{3}}{(5-\sqrt{3})(5+\sqrt{3})} = \frac{5+\sqrt{3}}{5^2 - (\sqrt{3})^2}

Step 5: Simplify the Expression

Now that we have simplified the denominator, we can simplify the expression.

5+352−(3)2=5+325−3\frac{5+\sqrt{3}}{5^2 - (\sqrt{3})^2} = \frac{5+\sqrt{3}}{25 - 3}

Step 6: Simplify the Expression

Now that we have simplified the denominator, we can simplify the expression.

5+325−3=5+322\frac{5+\sqrt{3}}{25 - 3} = \frac{5+\sqrt{3}}{22}

Conclusion

In this article, we have explored how to rationalize the denominator of the expression 15−3\frac{1}{5-\sqrt{3}} and simplify it to its simplest form. We have followed a step-by-step guide to rationalize the denominator, and we have used the difference of squares formula to simplify the denominator. By rationalizing the denominator, we can eliminate the square root in the denominator, making it easier to work with the expression.

Final Answer

The final answer is 5+322\boxed{\frac{5+\sqrt{3}}{22}}.

Common Mistakes to Avoid

When rationalizing the denominator, it's essential to avoid common mistakes. Here are some common mistakes to avoid:

  • Not multiplying the numerator and denominator by the conjugate: When rationalizing the denominator, it's essential to multiply the numerator and denominator by the conjugate of the denominator.
  • Not simplifying the denominator: When simplifying the denominator, it's essential to use the difference of squares formula to simplify the expression.
  • Not checking the final answer: When simplifying the expression, it's essential to check the final answer to ensure that it's correct.

Real-World Applications

Rationalizing the denominator has many real-world applications. Here are some examples:

  • Engineering: Rationalizing the denominator is used in engineering to simplify expressions that involve square roots in the denominator.
  • Physics: Rationalizing the denominator is used in physics to simplify expressions that involve square roots in the denominator.
  • Computer Science: Rationalizing the denominator is used in computer science to simplify expressions that involve square roots in the denominator.

Conclusion

In conclusion, rationalizing the denominator is a technique used to simplify expressions that involve square roots in the denominator. By following a step-by-step guide, we can rationalize the denominator and simplify the expression to its simplest form. This technique has many real-world applications and is essential in algebra and other advanced math topics.

Introduction

Rationalizing the denominator is a technique used to simplify expressions that involve square roots in the denominator. In our previous article, we explored how to rationalize the denominator of the expression 15−3\frac{1}{5-\sqrt{3}} and simplify it to its simplest form. In this article, we'll answer some frequently asked questions about rationalizing the denominator.

Q: What is rationalizing the denominator?

A: Rationalizing the denominator is a technique used to simplify expressions that involve square roots in the denominator. It involves multiplying the numerator and denominator by a value that will cancel out the square root in the denominator.

Q: Why is rationalizing the denominator important?

A: Rationalizing the denominator is important because it allows us to simplify expressions that involve square roots in the denominator. This technique is essential in algebra and is used to simplify expressions that involve square roots.

Q: How do I rationalize the denominator?

A: To rationalize the denominator, you need to follow these steps:

  1. Identify the value to multiply: You need to multiply the numerator and denominator by the conjugate of the denominator.
  2. Multiply the numerator and denominator: Multiply the numerator and denominator by the conjugate of the denominator.
  3. Simplify the expression: Simplify the expression by using the difference of squares formula to simplify the denominator.

Q: What is the conjugate of the denominator?

A: The conjugate of the denominator is the value that, when multiplied by the denominator, will cancel out the square root in the denominator. For example, if the denominator is 5−35-\sqrt{3}, the conjugate is 5+35+\sqrt{3}.

Q: How do I simplify the denominator?

A: To simplify the denominator, you need to use the difference of squares formula, which states that (a−b)(a+b)=a2−b2(a-b)(a+b) = a^2 - b^2. This formula allows you to simplify the denominator by canceling out the square root.

Q: What are some common mistakes to avoid when rationalizing the denominator?

A: Some common mistakes to avoid when rationalizing the denominator include:

  • Not multiplying the numerator and denominator by the conjugate
  • Not simplifying the denominator
  • Not checking the final answer

Q: What are some real-world applications of rationalizing the denominator?

A: Rationalizing the denominator has many real-world applications, including:

  • Engineering: Rationalizing the denominator is used in engineering to simplify expressions that involve square roots in the denominator.
  • Physics: Rationalizing the denominator is used in physics to simplify expressions that involve square roots in the denominator.
  • Computer Science: Rationalizing the denominator is used in computer science to simplify expressions that involve square roots in the denominator.

Q: Can I use a calculator to rationalize the denominator?

A: Yes, you can use a calculator to rationalize the denominator. However, it's essential to understand the concept of rationalizing the denominator and how to do it manually.

Q: How do I know if I've rationalized the denominator correctly?

A: To know if you've rationalized the denominator correctly, you need to check the final answer to ensure that it's correct. You can do this by plugging the final answer into the original expression and simplifying it.

Conclusion

In conclusion, rationalizing the denominator is a technique used to simplify expressions that involve square roots in the denominator. By following a step-by-step guide, we can rationalize the denominator and simplify the expression to its simplest form. This technique has many real-world applications and is essential in algebra and other advanced math topics.

Final Answer

The final answer is 5+322\boxed{\frac{5+\sqrt{3}}{22}}.

Common Mistakes to Avoid

When rationalizing the denominator, it's essential to avoid common mistakes. Here are some common mistakes to avoid:

  • Not multiplying the numerator and denominator by the conjugate: When rationalizing the denominator, it's essential to multiply the numerator and denominator by the conjugate of the denominator.
  • Not simplifying the denominator: When simplifying the denominator, it's essential to use the difference of squares formula to simplify the expression.
  • Not checking the final answer: When simplifying the expression, it's essential to check the final answer to ensure that it's correct.

Real-World Applications

Rationalizing the denominator has many real-world applications. Here are some examples:

  • Engineering: Rationalizing the denominator is used in engineering to simplify expressions that involve square roots in the denominator.
  • Physics: Rationalizing the denominator is used in physics to simplify expressions that involve square roots in the denominator.
  • Computer Science: Rationalizing the denominator is used in computer science to simplify expressions that involve square roots in the denominator.

Conclusion

In conclusion, rationalizing the denominator is a technique used to simplify expressions that involve square roots in the denominator. By following a step-by-step guide, we can rationalize the denominator and simplify the expression to its simplest form. This technique has many real-world applications and is essential in algebra and other advanced math topics.