Distribute And Simplify The Following Radicals:$\sqrt{12} \cdot (-1+\sqrt{5}$\]A. \[$-4 \sqrt{3}\$\]B. \[$-2 \sqrt{3} + 2 \sqrt{15}\$\]C. \[$4 \sqrt{3}\$\]D. \[$6 \sqrt{3}\$\]

Introduction

Radicals are an essential part of mathematics, and understanding how to distribute and simplify them is crucial for solving complex equations and expressions. In this article, we will focus on distributing and simplifying the radical $\sqrt{12} \cdot (-1+\sqrt{5})$. We will explore the different methods and techniques used to simplify radicals and provide step-by-step solutions to help you understand the process.

Understanding Radicals

A radical is a mathematical expression that represents the square root of a number. It is denoted by the symbol $\sqrt{}$. For example, $\sqrt{16}$ represents the square root of 16, which is equal to 4. Radicals can be simplified by finding the square root of the number inside the radical sign.

Distributing Radicals

Distributing radicals involves multiplying the radical by each term inside the parentheses. In the given expression, we have $\sqrt{12} \cdot (-1+\sqrt{5})$. To distribute the radical, we need to multiply $\sqrt{12}$ by each term inside the parentheses.

Step 1: Multiply $\sqrt{12}$ by $-1$

To multiply $\sqrt{12}$ by $-1$, we need to multiply the square root of 12 by $-1$. This can be written as:

$$\sqrt{12} \cdot (-1) = -\sqrt{12}$$

Step 2: Multiply $\sqrt{12}$ by $\sqrt{5}$

To multiply $\sqrt{12}$ by $\sqrt{5}$, we need to multiply the square root of 12 by the square root of 5. This can be written as:

$$\sqrt{12} \cdot \sqrt{5} = \sqrt{12 \cdot 5} = \sqrt{60}$$

Simplifying Radicals

Now that we have distributed the radical, we need to simplify the expression. To simplify the expression, we need to find the square root of the numbers inside the radical signs.

Step 1: Simplify $\sqrt{12}$

To simplify $\sqrt{12}$, we need to find the square root of 12. This can be written as:

$$\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}$$

Step 2: Simplify $\sqrt{60}$

To simplify $\sqrt{60}$, we need to find the square root of 60. This can be written as:

$$\sqrt{60} = \sqrt{4 \cdot 15} = 2\sqrt{15}$$

Combining the Terms

Now that we have simplified the radicals, we can combine the terms. The expression can be written as:

$$-2\sqrt{3} + 2\sqrt{15}$$

Conclusion

In this article, we have distributed and simplified the radical $\sqrt{12} \cdot (-1+\sqrt{5})$. We have used the distributive property to multiply the radical by each term inside the parentheses and then simplified the expression by finding the square root of the numbers inside the radical signs. The final expression is $-2\sqrt{3} + 2\sqrt{15}$.

Answer

The correct answer is:

• B. $-2 \sqrt{3} + 2 \sqrt{15}$

Final Thoughts

Introduction

In our previous article, we discussed how to distribute and simplify the radical $\sqrt{12} \cdot (-1+\sqrt{5})$. We provided step-by-step solutions and explained the concepts in detail. However, we know that practice makes perfect, and the best way to learn is by asking questions and getting answers. In this article, we will provide a Q&A guide to help you understand the concepts of distributing and simplifying radicals.

Q1: What is the distributive property of radicals?

A1: The distributive property of radicals states that we can multiply a radical by each term inside the parentheses. This means that we can distribute the radical to each term inside the parentheses and then simplify the expression.

Q2: How do I distribute a radical to a binomial?

A2: To distribute a radical to a binomial, we need to multiply the radical by each term inside the parentheses. For example, if we have $\sqrt{12} \cdot (-1+\sqrt{5})$, we need to multiply $\sqrt{12}$ by $-1$ and then multiply $\sqrt{12}$ by $\sqrt{5}$.

Q3: How do I simplify a radical?

A3: To simplify a radical, we need to find the square root of the number inside the radical sign. For example, if we have $\sqrt{12}$, we can simplify it by finding the square root of 12, which is equal to $2\sqrt{3}$.

Q4: What is the difference between a rational and an irrational number?

A4: A rational number is a number that can be expressed as a fraction, i.e., it can be written in the form $\frac{a}{b}$, where $a$ and $b$ are integers. An irrational number, on the other hand, is a number that cannot be expressed as a fraction. For example, $\sqrt{2}$ is an irrational number because it cannot be expressed as a fraction.

Q5: How do I know if a number is rational or irrational?

A5: To determine if a number is rational or irrational, we need to check if it can be expressed as a fraction. If it can be expressed as a fraction, then it is a rational number. If it cannot be expressed as a fraction, then it is an irrational number.

Q6: What is the importance of simplifying radicals?

A6: Simplifying radicals is important because it helps us to simplify complex expressions and make them easier to work with. By simplifying radicals, we can reduce the complexity of an expression and make it easier to solve.

Q7: How do I combine like terms after simplifying radicals?

A7: To combine like terms after simplifying radicals, we need to add or subtract the coefficients of the like terms. For example, if we have $2\sqrt{3} + 3\sqrt{3}$, we can combine the like terms by adding the coefficients, which gives us $5\sqrt{3}$.

Q8: What are some common mistakes to avoid when simplifying radicals?

A8: Some common mistakes to avoid when simplifying radicals include:

• Not simplifying the radical completely
• Not combining like terms
• Not checking if the number is rational or irrational
• Not using the distributive property correctly

Conclusion

In this article, we have provided a Q&A guide to help you understand the concepts of distributing and simplifying radicals. We have answered common questions and provided examples to help you understand the concepts. By following the steps outlined in this article, you can simplify radicals and solve complex expressions with ease.

Final Thoughts

Distributing and simplifying radicals is an essential part of mathematics, and understanding how to do it is crucial for solving complex equations and expressions. By practicing and asking questions, you can improve your skills and become more confident in your ability to simplify radicals. Remember to always simplify radicals by finding the square root of the numbers inside the radical signs and to combine the terms to get the final expression.