To Rationalize The Denominator Of 5 − 7 9 − 14 \frac{5-\sqrt{7}}{9-\sqrt{14}} 9 − 14 ​ 5 − 7 ​ ​ , You Should Multiply The Expression By Which Fraction?A. 5 + 7 9 − 14 \frac{5+\sqrt{7}}{9-\sqrt{14}} 9 − 14 ​ 5 + 7 ​ ​ B. 9 − 14 9 − 14 \frac{9-\sqrt{14}}{9-\sqrt{14}} 9 − 14 ​ 9 − 14 ​ ​ C.

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Understanding Rationalizing the Denominator

Rationalizing the denominator is a process used in algebra to eliminate any radical expressions in the denominator of a fraction. This is particularly important when dealing with expressions that involve square roots, as it allows us to simplify and manipulate the expression more easily. In this article, we will explore how to rationalize the denominator of a given expression, using the example of 5−79−14\frac{5-\sqrt{7}}{9-\sqrt{14}}.

Why Rationalize the Denominator?

Rationalizing the denominator is essential in mathematics, particularly in algebra and calculus. It allows us to simplify complex expressions, making them easier to work with and understand. By eliminating the radical in the denominator, we can perform operations such as addition, subtraction, multiplication, and division more easily. Additionally, rationalizing the denominator is a crucial step in solving equations and inequalities that involve radical expressions.

How to Rationalize the Denominator

To rationalize the denominator of an expression, we need to multiply the expression by a fraction that will eliminate the radical in the denominator. The key is to find a fraction that, when multiplied, will result in the radical being eliminated. In the case of the expression 5−79−14\frac{5-\sqrt{7}}{9-\sqrt{14}}, we need to find a fraction that will eliminate the radical in the denominator.

Step 1: Identify the Radical in the Denominator

The radical in the denominator is 14\sqrt{14}. To eliminate this radical, we need to find a fraction that will result in the radical being eliminated when multiplied.

Step 2: Find the Conjugate of the Denominator

The conjugate of the denominator is 9+149+\sqrt{14}. The conjugate is a key concept in rationalizing the denominator, as it allows us to eliminate the radical in the denominator.

Step 3: Multiply the Expression by the Conjugate

To rationalize the denominator, we need to multiply the expression by the conjugate of the denominator. In this case, we multiply the expression 5−79−14\frac{5-\sqrt{7}}{9-\sqrt{14}} by 9+149+14\frac{9+\sqrt{14}}{9+\sqrt{14}}.

Step 4: Simplify the Expression

When we multiply the expression by the conjugate, we get:

5−79−14⋅9+149+14=(5−7)(9+14)(9−14)(9+14)\frac{5-\sqrt{7}}{9-\sqrt{14}} \cdot \frac{9+\sqrt{14}}{9+\sqrt{14}} = \frac{(5-\sqrt{7})(9+\sqrt{14})}{(9-\sqrt{14})(9+\sqrt{14})}

Step 5: Simplify the Numerator and Denominator

To simplify the expression, we need to multiply out the numerator and denominator.

(5−7)(9+14)(9−14)(9+14)=45+514−97−9881−14\frac{(5-\sqrt{7})(9+\sqrt{14})}{(9-\sqrt{14})(9+\sqrt{14})} = \frac{45+5\sqrt{14}-9\sqrt{7}-\sqrt{98}}{81-14}

Step 6: Simplify the Expression Further

We can simplify the expression further by combining like terms.

45+514−97−9867=45+514−97−7267\frac{45+5\sqrt{14}-9\sqrt{7}-\sqrt{98}}{67} = \frac{45+5\sqrt{14}-9\sqrt{7}-7\sqrt{2}}{67}

Conclusion

Rationalizing the denominator is an essential process in mathematics, particularly in algebra and calculus. By following the steps outlined in this article, we can rationalize the denominator of a given expression, making it easier to work with and understand. In the case of the expression 5−79−14\frac{5-\sqrt{7}}{9-\sqrt{14}}, we multiplied the expression by the conjugate of the denominator to eliminate the radical in the denominator.

Answer

To rationalize the denominator of 5−79−14\frac{5-\sqrt{7}}{9-\sqrt{14}}, you should multiply the expression by 9+149+14\frac{9+\sqrt{14}}{9+\sqrt{14}}.

Discussion

Rationalizing the denominator is a crucial concept in mathematics, and it is essential to understand how to do it correctly. By following the steps outlined in this article, you can rationalize the denominator of any expression that involves a radical in the denominator. If you have any questions or need further clarification, please don't hesitate to ask.

Example Problems

  1. Rationalize the denominator of 3−25−3\frac{3-\sqrt{2}}{5-\sqrt{3}}.
  2. Rationalize the denominator of 2+53−2\frac{2+\sqrt{5}}{3-\sqrt{2}}.
  3. Rationalize the denominator of 4−62−3\frac{4-\sqrt{6}}{2-\sqrt{3}}.

Solutions

  1. Multiply the expression by 5+35+3\frac{5+\sqrt{3}}{5+\sqrt{3}}.
  2. Multiply the expression by 3+23+2\frac{3+\sqrt{2}}{3+\sqrt{2}}.
  3. Multiply the expression by 2+32+3\frac{2+\sqrt{3}}{2+\sqrt{3}}.

Conclusion

Rationalizing the denominator is an essential process in mathematics, and it is crucial to understand how to do it correctly. By following the steps outlined in this article, you can rationalize the denominator of any expression that involves a radical in the denominator. If you have any questions or need further clarification, please don't hesitate to ask.

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Frequently Asked Questions

Q: What is rationalizing the denominator?

A: Rationalizing the denominator is a process used in algebra to eliminate any radical expressions in the denominator of a fraction. This is particularly important when dealing with expressions that involve square roots, as it allows us to simplify and manipulate the expression more easily.

Q: Why is rationalizing the denominator important?

A: Rationalizing the denominator is essential in mathematics, particularly in algebra and calculus. It allows us to simplify complex expressions, making them easier to work with and understand. By eliminating the radical in the denominator, we can perform operations such as addition, subtraction, multiplication, and division more easily.

Q: How do I rationalize the denominator of a fraction?

A: To rationalize the denominator of a fraction, you need to multiply the expression by a fraction that will eliminate the radical in the denominator. The key is to find a fraction that, when multiplied, will result in the radical being eliminated.

Q: What is the conjugate of a denominator?

A: The conjugate of a denominator is a key concept in rationalizing the denominator. It is a fraction that, when multiplied, will eliminate the radical in the denominator. For example, if the denominator is a−ba-b, the conjugate is a+ba+b.

Q: How do I find the conjugate of a denominator?

A: To find the conjugate of a denominator, you need to change the sign of the radical in the denominator. For example, if the denominator is a−ba-\sqrt{b}, the conjugate is a+ba+\sqrt{b}.

Q: Can I rationalize the denominator of a fraction with a negative exponent?

A: Yes, you can rationalize the denominator of a fraction with a negative exponent. To do this, you need to multiply the expression by a fraction that will eliminate the radical in the denominator, and then simplify the resulting expression.

Q: How do I rationalize the denominator of a fraction with a complex number in the denominator?

A: To rationalize the denominator of a fraction with a complex number in the denominator, you need to multiply the expression by a fraction that will eliminate the radical in the denominator, and then simplify the resulting expression.

Q: Can I rationalize the denominator of a fraction with a variable in the denominator?

A: Yes, you can rationalize the denominator of a fraction with a variable in the denominator. To do this, you need to multiply the expression by a fraction that will eliminate the radical in the denominator, and then simplify the resulting expression.

Q: How do I rationalize the denominator of a fraction with a radical in the numerator?

A: To rationalize the denominator of a fraction with a radical in the numerator, you need to multiply the expression by a fraction that will eliminate the radical in the numerator and denominator, and then simplify the resulting expression.

Q: Can I rationalize the denominator of a fraction with a negative number in the denominator?

A: Yes, you can rationalize the denominator of a fraction with a negative number in the denominator. To do this, you need to multiply the expression by a fraction that will eliminate the radical in the denominator, and then simplify the resulting expression.

Q: How do I rationalize the denominator of a fraction with a decimal in the denominator?

A: To rationalize the denominator of a fraction with a decimal in the denominator, you need to multiply the expression by a fraction that will eliminate the decimal in the denominator, and then simplify the resulting expression.

Q: Can I rationalize the denominator of a fraction with a mixed number in the denominator?

A: Yes, you can rationalize the denominator of a fraction with a mixed number in the denominator. To do this, you need to multiply the expression by a fraction that will eliminate the radical in the denominator, and then simplify the resulting expression.

Q: How do I rationalize the denominator of a fraction with a fraction in the denominator?

A: To rationalize the denominator of a fraction with a fraction in the denominator, you need to multiply the expression by a fraction that will eliminate the radical in the denominator, and then simplify the resulting expression.

Q: Can I rationalize the denominator of a fraction with a negative fraction in the denominator?

A: Yes, you can rationalize the denominator of a fraction with a negative fraction in the denominator. To do this, you need to multiply the expression by a fraction that will eliminate the radical in the denominator, and then simplify the resulting expression.

Q: How do I rationalize the denominator of a fraction with a complex fraction in the denominator?

A: To rationalize the denominator of a fraction with a complex fraction in the denominator, you need to multiply the expression by a fraction that will eliminate the radical in the denominator, and then simplify the resulting expression.

Q: Can I rationalize the denominator of a fraction with a variable in the numerator and denominator?

A: Yes, you can rationalize the denominator of a fraction with a variable in the numerator and denominator. To do this, you need to multiply the expression by a fraction that will eliminate the radical in the denominator, and then simplify the resulting expression.

Q: How do I rationalize the denominator of a fraction with a radical in the numerator and denominator?

A: To rationalize the denominator of a fraction with a radical in the numerator and denominator, you need to multiply the expression by a fraction that will eliminate the radical in the numerator and denominator, and then simplify the resulting expression.

Q: Can I rationalize the denominator of a fraction with a negative number in the numerator and denominator?

A: Yes, you can rationalize the denominator of a fraction with a negative number in the numerator and denominator. To do this, you need to multiply the expression by a fraction that will eliminate the radical in the denominator, and then simplify the resulting expression.

Q: How do I rationalize the denominator of a fraction with a decimal in the numerator and denominator?

A: To rationalize the denominator of a fraction with a decimal in the numerator and denominator, you need to multiply the expression by a fraction that will eliminate the decimal in the denominator, and then simplify the resulting expression.

Q: Can I rationalize the denominator of a fraction with a mixed number in the numerator and denominator?

A: Yes, you can rationalize the denominator of a fraction with a mixed number in the numerator and denominator. To do this, you need to multiply the expression by a fraction that will eliminate the radical in the denominator, and then simplify the resulting expression.

Q: How do I rationalize the denominator of a fraction with a fraction in the numerator and denominator?

A: To rationalize the denominator of a fraction with a fraction in the numerator and denominator, you need to multiply the expression by a fraction that will eliminate the radical in the denominator, and then simplify the resulting expression.

Q: Can I rationalize the denominator of a fraction with a negative fraction in the numerator and denominator?

A: Yes, you can rationalize the denominator of a fraction with a negative fraction in the numerator and denominator. To do this, you need to multiply the expression by a fraction that will eliminate the radical in the denominator, and then simplify the resulting expression.

Q: How do I rationalize the denominator of a fraction with a complex fraction in the numerator and denominator?

A: To rationalize the denominator of a fraction with a complex fraction in the numerator and denominator, you need to multiply the expression by a fraction that will eliminate the radical in the denominator, and then simplify the resulting expression.

Q: Can I rationalize the denominator of a fraction with a variable in the numerator and a radical in the denominator?

A: Yes, you can rationalize the denominator of a fraction with a variable in the numerator and a radical in the denominator. To do this, you need to multiply the expression by a fraction that will eliminate the radical in the denominator, and then simplify the resulting expression.

Q: How do I rationalize the denominator of a fraction with a radical in the numerator and a variable in the denominator?

A: To rationalize the denominator of a fraction with a radical in the numerator and a variable in the denominator, you need to multiply the expression by a fraction that will eliminate the radical in the numerator and denominator, and then simplify the resulting expression.

Q: Can I rationalize the denominator of a fraction with a negative number in the numerator and a radical in the denominator?

A: Yes, you can rationalize the denominator of a fraction with a negative number in the numerator and a radical in the denominator. To do this, you need to multiply the expression by a fraction that will eliminate the radical in the denominator, and then simplify the resulting expression.

Q: How do I rationalize the denominator of a fraction with a decimal in the numerator and a radical in the denominator?

A: To rationalize the denominator of a fraction with a decimal in the numerator and a radical in the denominator, you need to multiply the expression by a fraction that will eliminate the decimal in the denominator, and then simplify the resulting expression.

Q: Can I rationalize the denominator of a fraction with a mixed number in the numerator and a radical in the denominator?

A: Yes,