Rationalize $\frac{8}{\sqrt{5}+\sqrt{7}}$.

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Introduction

When dealing with expressions that contain radicals in the denominator, it's essential to rationalize the denominator to simplify the expression and make it easier to work with. Rationalizing the denominator involves getting rid of the radical in the denominator by multiplying the numerator and denominator by a suitable expression. In this article, we'll focus on rationalizing the denominator of the expression 85+7\frac{8}{\sqrt{5}+\sqrt{7}}.

What is Rationalizing the Denominator?

Rationalizing the denominator is a process of eliminating radicals from the denominator of a fraction. This is done by multiplying the numerator and denominator by a suitable expression that will eliminate the radical in the denominator. The goal is to simplify the expression and make it easier to work with.

Why is Rationalizing the Denominator Important?

Rationalizing the denominator is an essential skill in mathematics, particularly when working with expressions that contain radicals. It's crucial to rationalize the denominator to:

  • Simplify expressions and make them easier to work with
  • Eliminate radicals from the denominator
  • Make it easier to perform operations such as addition, subtraction, multiplication, and division
  • Prepare expressions for further simplification or manipulation

How to Rationalize the Denominator

To rationalize the denominator of the expression 85+7\frac{8}{\sqrt{5}+\sqrt{7}}, we'll follow these steps:

Step 1: Identify the Radical in the Denominator

The radical in the denominator is 5+7\sqrt{5}+\sqrt{7}.

Step 2: Determine the Suitable Expression to Multiply

To eliminate the radical in the denominator, we need to multiply the numerator and denominator by a suitable expression that will eliminate the radical. In this case, we can multiply the numerator and denominator by the conjugate of the denominator, which is 5−7\sqrt{5}-\sqrt{7}.

Step 3: Multiply the Numerator and Denominator

We'll multiply the numerator and denominator by the conjugate of the denominator:

85+7⋅5−75−7\frac{8}{\sqrt{5}+\sqrt{7}} \cdot \frac{\sqrt{5}-\sqrt{7}}{\sqrt{5}-\sqrt{7}}

Step 4: Simplify the Expression

We'll simplify the expression by multiplying the numerator and denominator:

8(5−7)(5+7)(5−7)\frac{8(\sqrt{5}-\sqrt{7})}{(\sqrt{5}+\sqrt{7})(\sqrt{5}-\sqrt{7})}

Step 5: Apply the Difference of Squares Formula

We'll apply the difference of squares formula to simplify the denominator:

(5+7)(5−7)=(5)2−(7)2=5−7=−2(\sqrt{5}+\sqrt{7})(\sqrt{5}-\sqrt{7}) = (\sqrt{5})^2 - (\sqrt{7})^2 = 5 - 7 = -2

Step 6: Simplify the Expression Further

We'll simplify the expression further by substituting the simplified denominator:

8(5−7)−2\frac{8(\sqrt{5}-\sqrt{7})}{-2}

Step 7: Final Simplification

We'll simplify the expression further by dividing the numerator by the denominator:

8(5−7)−2=−4(5−7)\frac{8(\sqrt{5}-\sqrt{7})}{-2} = -4(\sqrt{5}-\sqrt{7})

Conclusion

Rationalizing the denominator of the expression 85+7\frac{8}{\sqrt{5}+\sqrt{7}} involves multiplying the numerator and denominator by the conjugate of the denominator. By following the steps outlined above, we can simplify the expression and eliminate the radical in the denominator. This is an essential skill in mathematics, particularly when working with expressions that contain radicals.

Example Problems

Problem 1

Rationalize the denominator of the expression 32+3\frac{3}{\sqrt{2}+\sqrt{3}}.

Solution

To rationalize the denominator, we'll multiply the numerator and denominator by the conjugate of the denominator, which is 2−3\sqrt{2}-\sqrt{3}.

32+3⋅2−32−3\frac{3}{\sqrt{2}+\sqrt{3}} \cdot \frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}-\sqrt{3}}

We'll simplify the expression by multiplying the numerator and denominator:

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

We'll apply the difference of squares formula to simplify the denominator:

(2+3)(2−3)=(2)2−(3)2=2−3=−1(\sqrt{2}+\sqrt{3})(\sqrt{2}-\sqrt{3}) = (\sqrt{2})^2 - (\sqrt{3})^2 = 2 - 3 = -1

We'll simplify the expression further by substituting the simplified denominator:

3(2−3)−1\frac{3(\sqrt{2}-\sqrt{3})}{-1}

We'll simplify the expression further by dividing the numerator by the denominator:

3(2−3)−1=−3(2−3)\frac{3(\sqrt{2}-\sqrt{3})}{-1} = -3(\sqrt{2}-\sqrt{3})

Problem 2

Rationalize the denominator of the expression 23−2\frac{2}{\sqrt{3}-\sqrt{2}}.

Solution

To rationalize the denominator, we'll multiply the numerator and denominator by the conjugate of the denominator, which is 3+2\sqrt{3}+\sqrt{2}.

23−2⋅3+23+2\frac{2}{\sqrt{3}-\sqrt{2}} \cdot \frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}+\sqrt{2}}

We'll simplify the expression by multiplying the numerator and denominator:

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

We'll apply the difference of squares formula to simplify the denominator:

(3−2)(3+2)=(3)2−(2)2=3−2=1(\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2}) = (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1

We'll simplify the expression further by substituting the simplified denominator:

2(3+2)1\frac{2(\sqrt{3}+\sqrt{2})}{1}

We'll simplify the expression further by dividing the numerator by the denominator:

2(3+2)1=2(3+2)\frac{2(\sqrt{3}+\sqrt{2})}{1} = 2(\sqrt{3}+\sqrt{2})

Final Thoughts

Rationalizing the denominator is an essential skill in mathematics, particularly when working with expressions that contain radicals. By following the steps outlined above, we can simplify expressions and eliminate radicals from the denominator. This is a crucial skill to master, and with practice, you'll become proficient in rationalizing denominators in no time.

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Introduction

Rationalizing the denominator is a crucial skill in mathematics, particularly when working with expressions that contain radicals. In our previous article, we provided a step-by-step guide on how to rationalize the denominator of the expression 85+7\frac{8}{\sqrt{5}+\sqrt{7}}. In this article, we'll answer some frequently asked questions about rationalizing the denominator.

Q&A

Q: What is rationalizing the denominator?

A: Rationalizing the denominator is a process of eliminating radicals from the denominator of a fraction. This is done by multiplying the numerator and denominator by a suitable expression that will eliminate the radical in the denominator.

Q: Why is rationalizing the denominator important?

A: Rationalizing the denominator is essential in mathematics, particularly when working with expressions that contain radicals. It's crucial to rationalize the denominator to simplify expressions, eliminate radicals from the denominator, and make it easier to perform operations such as addition, subtraction, multiplication, and division.

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

A: To rationalize the denominator of a fraction, you'll need to multiply the numerator and denominator by a suitable expression that will eliminate the radical in the denominator. This expression is usually the conjugate of the denominator.

Q: What is the conjugate of a denominator?

A: The conjugate of a denominator is an expression that, when multiplied by the denominator, will eliminate the radical. For example, the conjugate of 5+7\sqrt{5}+\sqrt{7} is 5−7\sqrt{5}-\sqrt{7}.

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

A: To find the conjugate of a denominator, you'll need to change the sign of the second term. For example, the conjugate of 5+7\sqrt{5}+\sqrt{7} is 5−7\sqrt{5}-\sqrt{7}.

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

A: Yes, you can rationalize the denominator of a fraction with a negative sign. To do this, you'll need to multiply the numerator and denominator by the conjugate of the denominator, just like you would with a positive sign.

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

A: Yes, you can rationalize the denominator of a fraction with a decimal or fraction in the denominator. To do this, you'll need to multiply the numerator and denominator by the conjugate of the denominator, just like you would with a radical in the denominator.

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

A: To check if you've rationalized the denominator correctly, you'll need to simplify the expression and make sure that the denominator is no longer a radical. If the denominator is still a radical, you'll need to multiply the numerator and denominator by the conjugate of the denominator again.

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

A: Yes, you can rationalize the denominator of a fraction with multiple radicals in the denominator. To do this, you'll need to multiply the numerator and denominator by the conjugate of the denominator, just like you would with a single radical in the denominator.

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

A: To rationalize the denominator of a fraction with a variable in the denominator, you'll need to multiply the numerator and denominator by the conjugate of the denominator, just like you would with a constant in the denominator.

Example Problems

Problem 1

Rationalize the denominator of the expression 32+3\frac{3}{\sqrt{2}+\sqrt{3}}.

Solution

To rationalize the denominator, we'll multiply the numerator and denominator by the conjugate of the denominator, which is 2−3\sqrt{2}-\sqrt{3}.

32+3⋅2−32−3\frac{3}{\sqrt{2}+\sqrt{3}} \cdot \frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}-\sqrt{3}}

We'll simplify the expression by multiplying the numerator and denominator:

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

We'll apply the difference of squares formula to simplify the denominator:

(2+3)(2−3)=(2)2−(3)2=2−3=−1(\sqrt{2}+\sqrt{3})(\sqrt{2}-\sqrt{3}) = (\sqrt{2})^2 - (\sqrt{3})^2 = 2 - 3 = -1

We'll simplify the expression further by substituting the simplified denominator:

3(2−3)−1\frac{3(\sqrt{2}-\sqrt{3})}{-1}

We'll simplify the expression further by dividing the numerator by the denominator:

3(2−3)−1=−3(2−3)\frac{3(\sqrt{2}-\sqrt{3})}{-1} = -3(\sqrt{2}-\sqrt{3})

Problem 2

Rationalize the denominator of the expression 23−2\frac{2}{\sqrt{3}-\sqrt{2}}.

Solution

To rationalize the denominator, we'll multiply the numerator and denominator by the conjugate of the denominator, which is 3+2\sqrt{3}+\sqrt{2}.

23−2⋅3+23+2\frac{2}{\sqrt{3}-\sqrt{2}} \cdot \frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}+\sqrt{2}}

We'll simplify the expression by multiplying the numerator and denominator:

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

We'll apply the difference of squares formula to simplify the denominator:

(3−2)(3+2)=(3)2−(2)2=3−2=1(\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2}) = (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1

We'll simplify the expression further by substituting the simplified denominator:

2(3+2)1\frac{2(\sqrt{3}+\sqrt{2})}{1}

We'll simplify the expression further by dividing the numerator by the denominator:

2(3+2)1=2(3+2)\frac{2(\sqrt{3}+\sqrt{2})}{1} = 2(\sqrt{3}+\sqrt{2})

Final Thoughts

Rationalizing the denominator is an essential skill in mathematics, particularly when working with expressions that contain radicals. By following the steps outlined above, you'll be able to rationalize the denominator of any fraction, regardless of the complexity of the expression. Remember to multiply the numerator and denominator by the conjugate of the denominator, and simplify the expression to ensure that the denominator is no longer a radical. With practice, you'll become proficient in rationalizing denominators in no time.