What Is This Expression In Simplified Form? 6 23 \frac{\sqrt{6}}{\sqrt{23}} 23 ​ 6 ​ ​ A. 14 3 7 \frac{14 \sqrt{3}}{7} 7 14 3 ​ ​ B. 138 23 \frac{\sqrt{138}}{\sqrt{23}} 23 ​ 138 ​ ​ C. 138 23 \frac{\sqrt{138}}{23} 23 138 ​ ​ D. 14 21 \frac{\sqrt{14}}{21} 21 14 ​ ​

Rationalizing the Denominator

When dealing with fractions that involve square roots, it's often necessary to rationalize the denominator. This process involves multiplying both the numerator and the denominator by a specific value to eliminate the square root from the denominator. In this case, we're given the expression $\frac{\sqrt{6}}{\sqrt{23}}$ and we need to simplify it.

Step 1: Multiply by the Conjugate

To rationalize the denominator, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $\sqrt{23}$ is also $\sqrt{23}$. By multiplying both the numerator and the denominator by $\sqrt{23}$, we get:

$$\frac{\sqrt{6}}{\sqrt{23}} \cdot \frac{\sqrt{23}}{\sqrt{23}} = \frac{\sqrt{6}\sqrt{23}}{\sqrt{23}\sqrt{23}}$$

Step 2: Simplify the Expression

Now that we've multiplied the numerator and the denominator by $\sqrt{23}$, we can simplify the expression. The denominator becomes $\sqrt{23}\sqrt{23}$, which is equal to $23$. The numerator becomes $\sqrt{6}\sqrt{23}$, which is equal to $\sqrt{138}$.

$$\frac{\sqrt{6}\sqrt{23}}{\sqrt{23}\sqrt{23}} = \frac{\sqrt{138}}{23}$$

Conclusion

Therefore, the simplified form of the expression $\frac{\sqrt{6}}{\sqrt{23}}$ is $\frac{\sqrt{138}}{23}$.

Comparison with Answer Choices

Let's compare our simplified expression with the answer choices:

• A. $\frac{14 \sqrt{3}}{7}$: This is not equal to our simplified expression.
• B. $\frac{\sqrt{138}}{\sqrt{23}}$: This is not equal to our simplified expression, as the denominator is still a square root.
• C. $\frac{\sqrt{138}}{23}$: This is equal to our simplified expression.
• D. $\frac{\sqrt{14}}{21}$: This is not equal to our simplified expression.

Final Answer

Q: What is rationalizing the denominator?

A: Rationalizing the denominator is a process of multiplying both the numerator and the denominator of a fraction by a specific value to eliminate the square root from the denominator.

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 from the denominator, which can make the expression more manageable.

Q: How do we rationalize the denominator?

A: To rationalize the denominator, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a square root is the same square root.

Q: What is the conjugate of a square root?

A: The conjugate of a square root is the same square root. For example, the conjugate of $\sqrt{23}$ is also $\sqrt{23}$.

Q: Can we rationalize the denominator of any expression?

A: Yes, we can rationalize the denominator of any expression that involves a square root in the denominator.

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

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

• Not multiplying both the numerator and the denominator by the conjugate of the denominator.
• Not simplifying the expression after rationalizing the denominator.
• Not checking if the expression can be simplified further.

Q: How do we know if an expression can be simplified further?

A: We can check if an expression can be simplified further by looking for any common factors in the numerator and the denominator. If there are any common factors, we can simplify the expression further.

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

A: Yes, we can rationalize the denominator of a fraction with a negative exponent. However, we need to be careful when multiplying the numerator and the denominator by the conjugate of the denominator.

Q: What are some real-world applications of rationalizing the denominator?

A: Rationalizing the denominator has many real-world applications, including:

• Simplifying expressions in algebra and calculus.
• Solving equations and inequalities.
• Working with fractions and decimals in finance and economics.
• Simplifying expressions in physics and engineering.

Conclusion

Rationalizing the denominator is an important concept in mathematics that helps us simplify expressions and make them easier to work with. By understanding how to rationalize the denominator, we can solve equations and inequalities, simplify expressions, and work with fractions and decimals in various fields.