Rationalize The Denominator And Simplify The Expression:1. $55 \cdot \frac{1}{\sqrt{3}}$2. 57 ⋅ 5 14 − 2 57 \cdot \frac{5}{\sqrt{14} - 2} 57 ⋅ 14 ​ − 2 5 ​

Introduction

Rationalizing the denominator is a crucial step in simplifying expressions involving square roots. It 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 will explore two examples of rationalizing the denominator and simplifying expressions.

Example 1: Rationalizing the Denominator in $55 \cdot \frac{1}{\sqrt{3}}$

To rationalize the denominator in the expression $55 \cdot \frac{1}{\sqrt{3}}$, we need to get rid of the square root in the denominator. We can do this by multiplying the numerator and denominator by $\sqrt{3}$.

55 \cdot \frac{1}{\sqrt{3}} = 55 \cdot \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}

By multiplying the numerator and denominator by $\sqrt{3}$, we get:

55 \cdot \frac{1}{\sqrt{3}} = 55 \cdot \frac{\sqrt{3}}{\sqrt{3} \cdot \sqrt{3}}

Since $\sqrt{3} \cdot \sqrt{3} = 3$, we can simplify the expression further:

55 \cdot \frac{1}{\sqrt{3}} = 55 \cdot \frac{\sqrt{3}}{3}

Therefore, the rationalized form of the expression $55 \cdot \frac{1}{\sqrt{3}}$ is $\frac{55\sqrt{3}}{3}$.

Example 2: Rationalizing the Denominator in $57 \cdot \frac{5}{\sqrt{14} - 2}$

To rationalize the denominator in the expression $57 \cdot \frac{5}{\sqrt{14} - 2}$, we need to get rid of the square root in the denominator. We can do this by multiplying the numerator and denominator by the conjugate of the denominator, which is $\sqrt{14} + 2$.

57 \cdot \frac{5}{\sqrt{14} - 2} = 57 \cdot \frac{5}{\sqrt{14} - 2} \cdot \frac{\sqrt{14} + 2}{\sqrt{14} + 2}

By multiplying the numerator and denominator by $\sqrt{14} + 2$, we get:

57 \cdot \frac{5}{\sqrt{14} - 2} = 57 \cdot \frac{5(\sqrt{14} + 2)}{(\sqrt{14} - 2)(\sqrt{14} + 2)}

Using the difference of squares formula, we can simplify the denominator:

(\sqrt{14} - 2)(\sqrt{14} + 2) = (\sqrt{14})^2 - 2^2

Since $(\sqrt{14})^2 = 14$ and $2^2 = 4$, we can simplify the expression further:

(\sqrt{14} - 2)(\sqrt{14} + 2) = 14 - 4

Therefore, the denominator simplifies to $10$.

57 \cdot \frac{5}{\sqrt{14} - 2} = 57 \cdot \frac{5(\sqrt{14} + 2)}{10}

By canceling out the common factor of $5$, we get:

57 \cdot \frac{5}{\sqrt{14} - 2} = 57 \cdot \frac{\sqrt{14} + 2}{2}

Therefore, the rationalized form of the expression $57 \cdot \frac{5}{\sqrt{14} - 2}$ is $\frac{57(\sqrt{14} + 2)}{2}$.

Conclusion

Rationalizing the denominator is a crucial step in simplifying expressions involving square roots. By multiplying the numerator and denominator by a cleverly chosen value, we can get rid of the square root in the denominator and simplify the expression. In this article, we explored two examples of rationalizing the denominator and simplifying expressions. By following the steps outlined in this article, you can rationalize the denominator and simplify expressions involving square roots.

Tips and Tricks

• When rationalizing the denominator, always multiply the numerator and denominator by the conjugate of the denominator.
• Use the difference of squares formula to simplify the denominator.
• Cancel out common factors in the numerator and denominator to simplify the expression.

Practice Problems

1. Rationalize the denominator in the expression $\frac{1}{\sqrt{2} + 1}$.
2. Simplify the expression $\frac{3}{\sqrt{5} - 2}$.
3. Rationalize the denominator in the expression $\frac{4}{\sqrt{3} - 1}$.

Answer Key

1. $\frac{\sqrt{2} - 1}{2 - 1} = \sqrt{2} - 1$
2. $\frac{3(\sqrt{5} + 2)}{5 - 4} = 3(\sqrt{5} + 2)$
3. $\frac{4(\sqrt{3} + 1)}{3 - 1} = 2(\sqrt{3} + 1)$
   Rationalizing the Denominator: Q&A =====================================

Q: What is rationalizing the denominator?

A: Rationalizing the denominator is a process of getting rid of the square root in the denominator of a fraction by multiplying the numerator and denominator by a cleverly chosen value.

Q: Why do we need to rationalize the denominator?

A: We need to rationalize the denominator to simplify the expression and make it easier to work with. Rationalizing the denominator helps to eliminate the square root in the denominator, which can make the expression more manageable.

Q: How do I rationalize the denominator?

A: To rationalize the denominator, you need to multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a binomial expression is obtained by changing the sign of the second term. For example, the conjugate of $a + b$ is $a - b$.

Q: What is the conjugate of a binomial expression?

A: The conjugate of a binomial expression is obtained by changing the sign of the second term. For example, the conjugate of $a + b$ is $a - b$, and the conjugate of $c - d$ is $c + d$.

Q: How do I simplify the denominator after rationalizing it?

A: After rationalizing the denominator, you can simplify it by using the difference of squares formula. The difference of squares formula states that $(a + b)(a - b) = a^2 - b^2$.

Q: What is the difference of squares formula?

A: The difference of squares formula states that $(a + b)(a - b) = a^2 - b^2$. This formula can be used to simplify the denominator after rationalizing it.

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 numerator and denominator by the conjugate of the denominator, just like you would with a positive number.

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

A: To rationalize the denominator of a fraction with a negative number in the denominator, you need to multiply the numerator and denominator by the conjugate of the denominator. For example, if the denominator is $-a + b$, you would multiply the numerator and denominator by $-a - b$.

Q: Can I rationalize the denominator of a fraction with a square root in the numerator?

A: Yes, you can rationalize the denominator of a fraction with a square root in the numerator. To do this, you need to multiply the numerator and denominator by the conjugate of the denominator.

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

A: To rationalize the denominator of a fraction with a square root in the numerator, you need to multiply the numerator and denominator by the conjugate of the denominator. For example, if the numerator is $\sqrt{a} + b$ and the denominator is $c + d$, you would multiply the numerator and denominator by $c - d$.

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

A: Yes, you can rationalize the denominator of a fraction with a cube root in the denominator. To do this, you need to multiply the numerator and denominator by the conjugate of the denominator.

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

A: To rationalize the denominator of a fraction with a cube root in the denominator, you need to multiply the numerator and denominator by the conjugate of the denominator. For example, if the denominator is $\sqrt[3]{a} + b$, you would multiply the numerator and denominator by $\sqrt[3]{a} - b$.

Q: Can I rationalize the denominator of a fraction with a square root and a cube root in the denominator?

A: Yes, you can rationalize the denominator of a fraction with a square root and a cube root in the denominator. To do this, you need to multiply the numerator and denominator by the conjugate of the denominator.

Q: How do I rationalize the denominator of a fraction with a square root and a cube root in the denominator?

A: To rationalize the denominator of a fraction with a square root and a cube root in the denominator, you need to multiply the numerator and denominator by the conjugate of the denominator. For example, if the denominator is $\sqrt{a} + \sqrt[3]{b}$, you would multiply the numerator and denominator by $\sqrt{a} - \sqrt[3]{b}$.

Conclusion

Rationalizing the denominator is a crucial step in simplifying expressions involving square roots and cube roots. By multiplying the numerator and denominator by a cleverly chosen value, you can get rid of the square root or cube root in the denominator and simplify the expression. In this article, we explored various questions and answers related to rationalizing the denominator. By following the steps outlined in this article, you can rationalize the denominator and simplify expressions involving square roots and cube roots.