Multiplying $\frac{3}{\sqrt{17}-\sqrt{2}}$ By Which Fraction Will Produce An Equivalent Fraction With A Rational Denominator?A. $\frac{\sqrt{17}-\sqrt{2}}{\sqrt{17}-\sqrt{2}}$ B.

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Introduction

When dealing with fractions that have irrational numbers in the denominator, it can be challenging to simplify them. One way to simplify such fractions is to multiply them by a specific fraction that will eliminate the irrational part from the denominator. In this case, we are given the fraction 317−2\frac{3}{\sqrt{17}-\sqrt{2}} and we need to find the fraction that, when multiplied by this given fraction, will produce an equivalent fraction with a rational denominator.

Understanding the Concept of Rational Denominators

A rational denominator is a denominator that can be expressed as a ratio of two integers, i.e., a fraction. In other words, a rational denominator is a denominator that can be simplified to a whole number or a fraction of whole numbers. To achieve a rational denominator, we need to eliminate the irrational part from the denominator.

The Multiplication Method

To eliminate the irrational part from the denominator, we can multiply the given fraction by a fraction that has the conjugate of the denominator. The conjugate of a binomial expression a−ba-b is a+ba+b. In this case, the conjugate of 17−2\sqrt{17}-\sqrt{2} is 17+2\sqrt{17}+\sqrt{2}.

Finding the Correct Fraction

To find the correct fraction that, when multiplied by the given fraction, will produce an equivalent fraction with a rational denominator, we need to multiply the given fraction by the conjugate of the denominator. The conjugate of 17−2\sqrt{17}-\sqrt{2} is 17+2\sqrt{17}+\sqrt{2}. Therefore, the correct fraction is 17+217+2\frac{\sqrt{17}+\sqrt{2}}{\sqrt{17}+\sqrt{2}}.

Multiplying the Fractions

Now, let's multiply the given fraction by the correct fraction:

317−2⋅17+217+2\frac{3}{\sqrt{17}-\sqrt{2}} \cdot \frac{\sqrt{17}+\sqrt{2}}{\sqrt{17}+\sqrt{2}}

Simplifying the Expression

To simplify the expression, we can multiply the numerators and the denominators separately:

3(17+2)(17−2)(17+2)\frac{3(\sqrt{17}+\sqrt{2})}{(\sqrt{17}-\sqrt{2})(\sqrt{17}+\sqrt{2})}

Applying the Difference of Squares Formula

The denominator can be simplified using the difference of squares formula:

(a−b)(a+b)=a2−b2(a-b)(a+b) = a^2 - b^2

In this case, a=17a = \sqrt{17} and b=2b = \sqrt{2}. Therefore, the denominator can be simplified as follows:

(17−2)(17+2)=172−22(\sqrt{17}-\sqrt{2})(\sqrt{17}+\sqrt{2}) = \sqrt{17}^2 - \sqrt{2}^2

Simplifying the Denominator

Now, let's simplify the denominator:

172−22=17−2=15\sqrt{17}^2 - \sqrt{2}^2 = 17 - 2 = 15

Simplifying the Expression

Now that the denominator is simplified, we can simplify the expression:

3(17+2)15\frac{3(\sqrt{17}+\sqrt{2})}{15}

Conclusion

In conclusion, multiplying the given fraction 317−2\frac{3}{\sqrt{17}-\sqrt{2}} by the fraction 17+217+2\frac{\sqrt{17}+\sqrt{2}}{\sqrt{17}+\sqrt{2}} will produce an equivalent fraction with a rational denominator. The resulting fraction is 3(17+2)15\frac{3(\sqrt{17}+\sqrt{2})}{15}.

Final Answer

The final answer is 17+217+2\boxed{\frac{\sqrt{17}+\sqrt{2}}{\sqrt{17}+\sqrt{2}}}.

Discussion

The concept of multiplying a fraction by its conjugate to eliminate the irrational part from the denominator is a powerful tool in algebra. This technique can be used to simplify complex fractions and make them easier to work with. In this case, we used the difference of squares formula to simplify the denominator and arrive at the final answer.

Example Problems

Here are a few example problems that demonstrate the concept of multiplying a fraction by its conjugate:

  • 23−2â‹…3+23+2=2(3+2)(3−2)(3+2)=2(3+2)3−2=2(3+2)\frac{2}{\sqrt{3}-\sqrt{2}} \cdot \frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}+\sqrt{2}} = \frac{2(\sqrt{3}+\sqrt{2})}{(\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2})} = \frac{2(\sqrt{3}+\sqrt{2})}{3-2} = 2(\sqrt{3}+\sqrt{2})
  • 45+3â‹…5−35−3=4(5−3)(5+3)(5−3)=4(5−3)5−3=4(5−3)\frac{4}{\sqrt{5}+\sqrt{3}} \cdot \frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}-\sqrt{3}} = \frac{4(\sqrt{5}-\sqrt{3})}{(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})} = \frac{4(\sqrt{5}-\sqrt{3})}{5-3} = 4(\sqrt{5}-\sqrt{3})

These example problems demonstrate how the concept of multiplying a fraction by its conjugate can be used to simplify complex fractions and make them easier to work with.

Applications

The concept of multiplying a fraction by its conjugate has many applications in algebra and other areas of mathematics. Here are a few examples:

  • Simplifying complex fractions: The concept of multiplying a fraction by its conjugate can be used to simplify complex fractions and make them easier to work with.
  • Rationalizing denominators: The concept of multiplying a fraction by its conjugate can be used to rationalize denominators and eliminate irrational parts from the denominator.
  • Solving equations: The concept of multiplying a fraction by its conjugate can be used to solve equations that involve complex fractions.

Conclusion

In conclusion, multiplying a fraction by its conjugate is a powerful tool in algebra that can be used to simplify complex fractions and make them easier to work with. The concept of multiplying a fraction by its conjugate has many applications in algebra and other areas of mathematics, and it is an important technique to know when working with complex fractions.

Q&A

Q: What is the concept of multiplying a fraction by its conjugate?

A: The concept of multiplying a fraction by its conjugate is a powerful tool in algebra that can be used to simplify complex fractions and make them easier to work with. The conjugate of a binomial expression a−ba-b is a+ba+b. When we multiply a fraction by its conjugate, we eliminate the irrational part from the denominator.

Q: Why do we need to multiply a fraction by its conjugate?

A: We need to multiply a fraction by its conjugate to eliminate the irrational part from the denominator. This makes the fraction easier to work with and simplifies the expression.

Q: How do we find the conjugate of a binomial expression?

A: The conjugate of a binomial expression a−ba-b is a+ba+b. To find the conjugate, we simply change the sign of the second term.

Q: What is the difference of squares formula?

A: The difference of squares formula is (a−b)(a+b)=a2−b2(a-b)(a+b) = a^2 - b^2. This formula can be used to simplify the denominator of a fraction.

Q: How do we apply the difference of squares formula?

A: To apply the difference of squares formula, we simply multiply the two binomials together and simplify the expression.

Q: What is the final answer to the problem?

A: The final answer to the problem is 3(17+2)15\frac{3(\sqrt{17}+\sqrt{2})}{15}.

Q: Can you provide more example problems that demonstrate the concept of multiplying a fraction by its conjugate?

A: Here are a few more example problems:

  • 23−2â‹…3+23+2=2(3+2)(3−2)(3+2)=2(3+2)3−2=2(3+2)\frac{2}{\sqrt{3}-\sqrt{2}} \cdot \frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}+\sqrt{2}} = \frac{2(\sqrt{3}+\sqrt{2})}{(\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2})} = \frac{2(\sqrt{3}+\sqrt{2})}{3-2} = 2(\sqrt{3}+\sqrt{2})
  • 45+3â‹…5−35−3=4(5−3)(5+3)(5−3)=4(5−3)5−3=4(5−3)\frac{4}{\sqrt{5}+\sqrt{3}} \cdot \frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}-\sqrt{3}} = \frac{4(\sqrt{5}-\sqrt{3})}{(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})} = \frac{4(\sqrt{5}-\sqrt{3})}{5-3} = 4(\sqrt{5}-\sqrt{3})

Q: What are some applications of the concept of multiplying a fraction by its conjugate?

A: The concept of multiplying a fraction by its conjugate has many applications in algebra and other areas of mathematics. Here are a few examples:

  • Simplifying complex fractions: The concept of multiplying a fraction by its conjugate can be used to simplify complex fractions and make them easier to work with.
  • Rationalizing denominators: The concept of multiplying a fraction by its conjugate can be used to rationalize denominators and eliminate irrational parts from the denominator.
  • Solving equations: The concept of multiplying a fraction by its conjugate can be used to solve equations that involve complex fractions.

Q: Is there a shortcut to multiplying a fraction by its conjugate?

A: Yes, there is a shortcut to multiplying a fraction by its conjugate. The shortcut is to simply multiply the numerator and denominator by the conjugate of the denominator.

Q: Can you provide a step-by-step guide to multiplying a fraction by its conjugate?

A: Here is a step-by-step guide to multiplying a fraction by its conjugate:

  1. Identify the conjugate of the denominator.
  2. Multiply the numerator and denominator by the conjugate of the denominator.
  3. Simplify the expression using the difference of squares formula.
  4. Write the final answer in simplest form.

Q: What are some common mistakes to avoid when multiplying a fraction by its conjugate?

A: Here are some common mistakes to avoid when multiplying a fraction by its conjugate:

  • Not identifying the conjugate of the denominator.
  • Not multiplying the numerator and denominator by the conjugate of the denominator.
  • Not simplifying the expression using the difference of squares formula.
  • Not writing the final answer in simplest form.

Q: Can you provide more resources for learning about multiplying a fraction by its conjugate?

A: Here are some additional resources for learning about multiplying a fraction by its conjugate:

  • Online tutorials and videos
  • Algebra textbooks and workbooks
  • Online forums and discussion groups
  • Math education websites and blogs

Q: Is there a way to check if the final answer is correct?

A: Yes, there are several ways to check if the final answer is correct:

  • Plug the final answer back into the original equation to see if it is true.
  • Simplify the final answer to see if it is in simplest form.
  • Check the final answer against a known solution or reference solution.
  • Use a calculator or computer program to check the final answer.