Rationalize The Denominator And Simplify: 14 153 \sqrt{\frac{14}{153}} 153 14 ​ ​ .

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Introduction

Rationalizing the denominator is a crucial step in simplifying complex fractions, especially when dealing with square roots. In this article, we will delve into the process of rationalizing the denominator and simplifying the given expression: 14153\sqrt{\frac{14}{153}}. We will explore the step-by-step process, provide examples, and offer tips for simplifying complex fractions.

Understanding Rationalizing the Denominator

Rationalizing the denominator involves eliminating any radical expressions in the denominator of a fraction. This is achieved by multiplying both the numerator and denominator by a suitable expression that will eliminate the radical. In the case of square roots, we can rationalize the denominator by multiplying both the numerator and denominator by the square root of the number in the denominator.

Step-by-Step Process

To rationalize the denominator of 14153\sqrt{\frac{14}{153}}, we will follow these steps:

Step 1: Identify the Radical Expression in the Denominator

The given expression has a square root in the denominator, which is 153\sqrt{153}. This is the radical expression that we need to eliminate.

Step 2: Multiply the Numerator and Denominator by the Square Root of the Denominator

To eliminate the radical in the denominator, we will multiply both the numerator and denominator by 153\sqrt{153}. This will give us:

14153⋅153153=14⋅153153⋅153\sqrt{\frac{14}{153}} \cdot \frac{\sqrt{153}}{\sqrt{153}} = \frac{\sqrt{14 \cdot 153}}{\sqrt{153 \cdot 153}}

Step 3: Simplify the Expression

Now that we have eliminated the radical in the denominator, we can simplify the expression. We can simplify the numerator and denominator separately:

14⋅153=2142\sqrt{14 \cdot 153} = \sqrt{2142}

153⋅153=23409\sqrt{153 \cdot 153} = \sqrt{23409}

Step 4: Simplify the Square Root in the Numerator

We can simplify the square root in the numerator by finding the largest perfect square that divides 2142. We can write 2142 as:

2142=2⋅1071=2⋅32⋅1192142 = 2 \cdot 1071 = 2 \cdot 3^2 \cdot 119

Since 3 is a perfect square, we can simplify the square root in the numerator:

2142=2⋅32⋅119=32⋅119\sqrt{2142} = \sqrt{2 \cdot 3^2 \cdot 119} = 3\sqrt{2 \cdot 119}

Step 5: Simplify the Square Root in the Denominator

We can simplify the square root in the denominator by finding the largest perfect square that divides 23409. We can write 23409 as:

23409=32⋅259323409 = 3^2 \cdot 2593

Since 3 is a perfect square, we can simplify the square root in the denominator:

23409=32593\sqrt{23409} = 3\sqrt{2593}

Step 6: Simplify the Expression

Now that we have simplified the numerator and denominator, we can simplify the expression:

32⋅11932593=2⋅1192593\frac{3\sqrt{2 \cdot 119}}{3\sqrt{2593}} = \frac{\sqrt{2 \cdot 119}}{\sqrt{2593}}

Step 7: Simplify the Expression Further

We can simplify the expression further by finding the largest perfect square that divides 2 and 119. We can write 2 as:

2=212 = 2^1

Since 2 is a perfect square, we can simplify the expression:

2⋅1192593=21⋅59.52593\frac{\sqrt{2 \cdot 119}}{\sqrt{2593}} = \frac{\sqrt{2^1 \cdot 59.5}}{\sqrt{2593}}

However, we cannot simplify the expression further since 59.5 is not a perfect square.

Conclusion

Rationalizing the denominator and simplifying the given expression 14153\sqrt{\frac{14}{153}} involves several steps. We identified the radical expression in the denominator, multiplied the numerator and denominator by the square root of the denominator, simplified the expression, and finally simplified the square root in the numerator and denominator. The final simplified expression is 2⋅1192593\frac{\sqrt{2 \cdot 119}}{\sqrt{2593}}. This process demonstrates the importance of rationalizing the denominator in simplifying complex fractions.

Tips and Tricks

  • When rationalizing the denominator, always multiply both the numerator and denominator by the square root of the denominator.
  • Simplify the expression by finding the largest perfect square that divides the numerator and denominator.
  • Use the properties of square roots to simplify the expression further.

Common Mistakes

  • Failing to multiply both the numerator and denominator by the square root of the denominator.
  • Not simplifying the expression by finding the largest perfect square that divides the numerator and denominator.
  • Not using the properties of square roots to simplify the expression further.

Real-World Applications

Rationalizing the denominator and simplifying complex fractions has numerous real-world applications in various fields, including:

  • Physics: Rationalizing the denominator is essential in calculating the energy of a particle in a potential well.
  • Engineering: Rationalizing the denominator is crucial in designing electrical circuits and calculating the impedance of a circuit.
  • Finance: Rationalizing the denominator is necessary in calculating the interest rate of a loan and the return on investment.

Conclusion

Introduction

Rationalizing the denominator is a crucial step in simplifying complex fractions, especially when dealing with square roots. In this article, we will answer some frequently asked questions about rationalizing the denominator and simplifying complex fractions.

Q: What is rationalizing the denominator?

A: Rationalizing the denominator involves eliminating any radical expressions in the denominator of a fraction. This is achieved by multiplying both the numerator and denominator by a suitable expression that will eliminate the radical.

Q: Why is rationalizing the denominator important?

A: Rationalizing the denominator is essential in simplifying complex fractions and making them more manageable. It helps to eliminate any radical expressions in the denominator, which can make the fraction more difficult to work with.

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

A: To rationalize the denominator of a fraction, you need to multiply both the numerator and denominator by the square root of the denominator. This will eliminate any radical expressions in the denominator.

Q: What is the difference between rationalizing the denominator and simplifying a fraction?

A: Rationalizing the denominator involves eliminating any radical expressions in the denominator, while simplifying a fraction involves reducing the fraction to its simplest form. While rationalizing the denominator is a crucial step in simplifying a fraction, it is not the same thing.

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. However, you need to be careful when multiplying the numerator and denominator by the square root of the denominator, as this can result in a negative number.

Q: How do I simplify a fraction with a radical in the denominator?

A: To simplify a fraction with a radical in the denominator, you need to rationalize the denominator by multiplying both the numerator and denominator by the square root of the denominator. Then, you can simplify the expression by finding the largest perfect square that divides the numerator and denominator.

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

A: Yes, you can use a calculator to rationalize the denominator of a fraction. However, it is always a good idea to check your work by hand to ensure that you have not made any mistakes.

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

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

  • Failing to multiply both the numerator and denominator by the square root of the denominator.
  • Not simplifying the expression by finding the largest perfect square that divides the numerator and denominator.
  • Not using the properties of square roots to simplify the expression further.

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

A: To check if you have rationalized the denominator correctly, you can multiply the numerator and denominator by the square root of the denominator and simplify the expression. If the resulting expression is a simplified fraction, then you have rationalized the denominator correctly.

Conclusion

Rationalizing the denominator is a crucial step in simplifying complex fractions, especially when dealing with square roots. By following the steps outlined in this article and using the properties of square roots, you can simplify complex fractions and make them more manageable. Remember to be careful when multiplying the numerator and denominator by the square root of the denominator, and to check your work by hand to ensure that you have not made any mistakes.

Additional Resources

  • For more information on rationalizing the denominator, check out the following resources:
  • Khan Academy: Rationalizing the Denominator
  • Mathway: Rationalizing the Denominator
  • Wolfram Alpha: Rationalizing the Denominator

Practice Problems

  • Rationalize the denominator of the following fractions:
  • 1625\sqrt{\frac{16}{25}}
  • 916\sqrt{\frac{9}{16}}
  • 2536\sqrt{\frac{25}{36}}
  • Simplify the following fractions:
  • 23\frac{\sqrt{2}}{\sqrt{3}}
  • 32\frac{\sqrt{3}}{\sqrt{2}}
  • 56\frac{\sqrt{5}}{\sqrt{6}}