Which Value Is Equivalent To $2 \sqrt{3} \cdot 4 \sqrt{15}$?A. $15 \sqrt{2}$B. \$24 \sqrt{5}$[/tex\]C. $11 \sqrt{5}$D. $4 \sqrt{2}$

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying radical expressions, with a focus on multiplying radicals. We will use the given expression $2 \sqrt{3} \cdot 4 \sqrt{15}$ as an example to demonstrate the steps involved.

Understanding Radical Expressions

A radical expression is a mathematical expression that contains a square root or a higher root of a number. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16.

Multiplying Radical Expressions

When multiplying radical expressions, we can combine the radicals by multiplying the numbers inside the radicals. This is known as the product rule for radicals.

The Product Rule for Radicals

The product rule for radicals states that:

$$\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$$

This means that when we multiply two radical expressions, we can combine them by multiplying the numbers inside the radicals.

Simplifying the Given Expression

Now, let's apply the product rule for radicals to the given expression $2 \sqrt{3} \cdot 4 \sqrt{15}$.

$$2 \sqrt{3} \cdot 4 \sqrt{15} = \sqrt{3} \cdot \sqrt{15} \cdot 2 \cdot 4$$

Using the product rule for radicals, we can combine the radicals by multiplying the numbers inside the radicals.

$$\sqrt{3} \cdot \sqrt{15} = \sqrt{3 \cdot 15} = \sqrt{45}$$

Now, we can simplify the expression further by factoring the number inside the radical.

$$\sqrt{45} = \sqrt{9 \cdot 5} = 3 \sqrt{5}$$

So, the simplified expression is:

$$3 \sqrt{5} \cdot 2 \cdot 4 = 24 \sqrt{5}$$

Conclusion

In this article, we have demonstrated the process of simplifying radical expressions by multiplying radicals. We have used the product rule for radicals to combine the radicals and simplify the expression. The final answer is:

$$24 \sqrt{5}$$

This is the correct answer, which corresponds to option B.

Answer Key

• A. $15 \sqrt{2}$: Incorrect
• B. $24 \sqrt{5}$: Correct
• C. $11 \sqrt{5}$: Incorrect
• D. $4 \sqrt{2}$: Incorrect

Final Thoughts

Q: What is the product rule for radicals?

A: The product rule for radicals states that:

$$\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$$

This means that when we multiply two radical expressions, we can combine them by multiplying the numbers inside the radicals.

Q: How do I simplify a radical expression by multiplying radicals?

A: To simplify a radical expression by multiplying radicals, follow these steps:

1. Multiply the numbers inside the radicals.
2. Combine the radicals by multiplying the numbers inside the radicals.
3. Simplify the expression further by factoring the number inside the radical.

Q: What is the difference between a radical expression and a rational expression?

A: A radical expression is a mathematical expression that contains a square root or a higher root of a number. A rational expression is a mathematical expression that contains a fraction with a polynomial in the numerator and a polynomial in the denominator.

Q: Can I simplify a radical expression by dividing radicals?

A: Yes, you can simplify a radical expression by dividing radicals. To do this, follow these steps:

1. Invert the divisor (i.e., flip the fraction).
2. Multiply the numerator and denominator by the conjugate of the denominator.
3. Simplify the expression further by factoring the number inside the radical.

Q: What is the conjugate of a radical expression?

A: The conjugate of a radical expression is a radical expression with the opposite sign in the denominator. For example, the conjugate of $\sqrt{a} + \sqrt{b}$ is $\sqrt{a} - \sqrt{b}$.

Q: Can I simplify a radical expression by adding or subtracting radicals?

A: Yes, you can simplify a radical expression by adding or subtracting radicals. To do this, follow these steps:

1. Combine the radicals by adding or subtracting the numbers inside the radicals.
2. Simplify the expression further by factoring the number inside the radical.

Q: What is the difference between a simplified radical expression and a simplified rational expression?

A: A simplified radical expression is a radical expression that has been simplified by combining the radicals and factoring the number inside the radical. A simplified rational expression is a rational expression that has been simplified by canceling out common factors in the numerator and denominator.

Q: Can I simplify a radical expression with a negative number inside the radical?

A: Yes, you can simplify a radical expression with a negative number inside the radical. To do this, follow these steps:

1. Factor the number inside the radical.
2. Simplify the expression further by combining the radicals and factoring the number inside the radical.

Q: What is the final answer to the given expression $2 \sqrt{3} \cdot 4 \sqrt{15}$?

A: The final answer to the given expression $2 \sqrt{3} \cdot 4 \sqrt{15}$ is:

$$24 \sqrt{5}$$

This is the correct answer, which corresponds to option B.

Conclusion

In this article, we have answered some of the most frequently asked questions about simplifying radical expressions. We have covered topics such as the product rule for radicals, simplifying radical expressions by multiplying radicals, and simplifying radical expressions by adding or subtracting radicals. We hope that this article has been helpful in clarifying any confusion you may have had about simplifying radical expressions.