Fill In The Blanks In The Sentence Below So That The Resulting Statement Is True.The Number 2 18 \frac{2}{\sqrt{18}} 18 ​ 2 ​ Can Be Rewritten Without A Radical In The Denominator By First Simplifying 18 \sqrt{18} 18 ​ To □ \square □ And Then

Understanding the Problem

The given statement is $\frac{2}{\sqrt{18}}$, and we are asked to rewrite it without a radical in the denominator. To achieve this, we need to simplify $\sqrt{18}$ to a value that does not contain a radical. This can be done by factoring the number inside the square root sign.

Factoring the Number Inside the Square Root

The number $18$ can be factored as $18 = 9 \times 2$. Since $9$ is a perfect square ($9 = 3^2$), we can rewrite $\sqrt{18}$ as $\sqrt{9 \times 2}$. Using the property of radicals that $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$, we can simplify $\sqrt{18}$ as $\sqrt{9} \times \sqrt{2}$.

Simplifying the Square Root of 9

Since $9$ is a perfect square ($9 = 3^2$), we can rewrite $\sqrt{9}$ as $3$. Therefore, $\sqrt{18}$ can be simplified as $3 \times \sqrt{2}$.

Rewriting the Original Statement

Now that we have simplified $\sqrt{18}$ to $3 \times \sqrt{2}$, we can rewrite the original statement as $\frac{2}{3 \times \sqrt{2}}$. To rationalize the denominator, we need to multiply both the numerator and the denominator by the conjugate of the denominator.

Rationalizing the Denominator

The conjugate of $3 \times \sqrt{2}$ is $3 \times \sqrt{2}$. Multiplying both the numerator and the denominator by $3 \times \sqrt{2}$, we get:

$$\frac{2}{3 \times \sqrt{2}} \times \frac{3 \times \sqrt{2}}{3 \times \sqrt{2}} = \frac{2 \times 3 \times \sqrt{2}}{3 \times 3 \times 2}$$

Simplifying the Expression

We can simplify the expression by canceling out common factors in the numerator and the denominator. The numerator and the denominator have a common factor of $3$, which can be canceled out:

$$\frac{2 \times 3 \times \sqrt{2}}{3 \times 3 \times 2} = \frac{2 \times \sqrt{2}}{3 \times 2}$$

Final Simplification

We can further simplify the expression by canceling out the common factor of $2$ in the numerator and the denominator:

$$\frac{2 \times \sqrt{2}}{3 \times 2} = \frac{\sqrt{2}}{3}$$

Conclusion

In this article, we have shown how to simplify the radical in the denominator of the expression $\frac{2}{\sqrt{18}}$. By factoring the number inside the square root sign and simplifying the square root of $9$, we were able to rewrite the original statement without a radical in the denominator. The final simplified expression is $\frac{\sqrt{2}}{3}$.

Step-by-Step Solution

1. Factor the number inside the square root sign: $18 = 9 \times 2$
2. Simplify the square root of $9$: $\sqrt{9} = 3$
3. Rewrite $\sqrt{18}$ as $3 \times \sqrt{2}$
4. Rewrite the original statement as $\frac{2}{3 \times \sqrt{2}}$
5. Rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator
6. Simplify the expression by canceling out common factors in the numerator and the denominator
7. Final simplification by canceling out the common factor of $2$ in the numerator and the denominator

Key Concepts

• Factoring the number inside the square root sign
• Simplifying the square root of a perfect square
• Rationalizing the denominator
• Simplifying expressions by canceling out common factors

Practice Problems

• Simplify the radical in the denominator of the expression $\frac{3}{\sqrt{24}}$
• Rewrite the original statement without a radical in the denominator: $\frac{4}{\sqrt{32}}$
• Simplify the expression: $\frac{5}{\sqrt{45}}$
  Frequently Asked Questions (FAQs) =====================================

Q: What is the process of simplifying radicals in the denominator?

A: The process of simplifying radicals in the denominator involves factoring the number inside the square root sign, simplifying the square root of a perfect square, and then rationalizing the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator.

Q: How do I simplify the square root of a perfect square?

A: To simplify the square root of a perfect square, you need to find the square root of the number inside the square root sign. For example, if you have $\sqrt{9}$, you can simplify it as $3$ because $3^2 = 9$.

Q: What is rationalizing the denominator?

A: Rationalizing the denominator is the process of getting rid of the radical in the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator. This is done to simplify the expression and make it easier to work with.

Q: How do I rationalize the denominator?

A: To rationalize the denominator, you need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a binomial expression $a + b$ is $a - b$. For example, if you have $\frac{2}{3 + \sqrt{2}}$, you can rationalize the denominator by multiplying both the numerator and the denominator by $3 - \sqrt{2}$.

Q: What is the conjugate of a binomial expression?

A: The conjugate of a binomial expression $a + b$ is $a - b$. For example, the conjugate of $3 + \sqrt{2}$ is $3 - \sqrt{2}$.

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

A: To simplify an expression with a radical in the denominator, you need to follow these steps:

1. Factor the number inside the square root sign.
2. Simplify the square root of a perfect square.
3. Rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator.
4. Simplify the expression by canceling out common factors in the numerator and the denominator.

Q: What are some common mistakes to avoid when simplifying radicals in the denominator?

A: Some common mistakes to avoid when simplifying radicals in the denominator include:

• Not factoring the number inside the square root sign.
• Not simplifying the square root of a perfect square.
• Not rationalizing the denominator.
• Not simplifying the expression by canceling out common factors in the numerator and the denominator.

Q: How do I know if I have simplified the expression correctly?

A: To know if you have simplified the expression correctly, you need to check if the denominator is no longer a radical. If the denominator is no longer a radical, then you have simplified the expression correctly.

Additional Resources

• For more information on simplifying radicals in the denominator, check out the following resources:

• Khan Academy: Simplifying Radicals
• Mathway: Simplifying Radicals
• Wolfram Alpha: Simplifying Radicals

Practice Problems

• Simplify the radical in the denominator of the expression $\frac{3}{\sqrt{24}}$
• Rewrite the original statement without a radical in the denominator: $\frac{4}{\sqrt{32}}$
• Simplify the expression: $\frac{5}{\sqrt{45}}$

Conclusion

Simplifying radicals in the denominator is an important concept in algebra that can be used to simplify expressions and make them easier to work with. By following the steps outlined in this article, you can simplify radicals in the denominator and become more confident in your math skills.