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

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Understanding Radicals and Their Operations

Radicals are mathematical expressions that involve the extraction of the root of a number. They are commonly denoted by the symbol \sqrt{}. In this article, we will focus on distributing and simplifying radicals, which is an essential skill in mathematics, particularly in algebra and geometry.

The Basics of Distributing Radicals

When distributing radicals, we need to multiply the radical expression by each term inside the parentheses. This involves applying the distributive property, which states that for any real numbers aa, bb, and cc, a(b+c)=ab+aca(b+c) = ab + ac. In the context of radicals, this means that we can distribute the radical expression to each term inside the parentheses.

Distributing Radicals: A Step-by-Step Example

Let's consider the expression 12⋅(−1+5)\sqrt{12} \cdot (-1+\sqrt{5}). To distribute the radical expression, we need to multiply the radical by each term inside the parentheses.

12⋅(−1+5)=12⋅(−1)+12⋅5\sqrt{12} \cdot (-1+\sqrt{5}) = \sqrt{12} \cdot (-1) + \sqrt{12} \cdot \sqrt{5}

Simplifying the Radical Expression

Now that we have distributed the radical expression, we need to simplify the resulting expression. To do this, we need to apply the rules of radicals, which state that for any real numbers aa and bb, ab=a⋅b\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}.

12⋅(−1)=−12\sqrt{12} \cdot (-1) = -\sqrt{12}

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

Simplifying the Radical Expression Further

We can simplify the radical expression further by factoring out the perfect square from the radicand. In this case, we can factor out 44 from 6060, which is a perfect square.

60=4⋅15=4⋅15=215\sqrt{60} = \sqrt{4 \cdot 15} = \sqrt{4} \cdot \sqrt{15} = 2\sqrt{15}

Combining the Simplified Expressions

Now that we have simplified the radical expression, we can combine the two simplified expressions to get the final result.

−12+215-\sqrt{12} + 2\sqrt{15}

Simplifying the Radical Expression Again

We can simplify the radical expression again by factoring out the perfect square from the radicand. In this case, we can factor out 44 from 1212, which is a perfect square.

−12=−4⋅3=−4⋅3=−23-\sqrt{12} = -\sqrt{4 \cdot 3} = -\sqrt{4} \cdot \sqrt{3} = -2\sqrt{3}

Combining the Simplified Expressions Again

Now that we have simplified the radical expression again, we can combine the two simplified expressions to get the final result.

−23+215-2\sqrt{3} + 2\sqrt{15}

Conclusion

In this article, we have learned how to distribute and simplify radicals. We have applied the distributive property to distribute the radical expression to each term inside the parentheses and then simplified the resulting expression by applying the rules of radicals. We have also factored out the perfect square from the radicand to simplify the radical expression further. Finally, we have combined the simplified expressions to get the final result.

Answer

The final answer is −23+215\boxed{-2 \sqrt{3} + 2 \sqrt{15}}.

Discussion

This problem requires the application of the distributive property and the rules of radicals to simplify the radical expression. The student needs to carefully distribute the radical expression to each term inside the parentheses and then simplify the resulting expression by applying the rules of radicals. The student also needs to factor out the perfect square from the radicand to simplify the radical expression further.

Key Takeaways

  • Distributing radicals involves applying the distributive property to multiply the radical expression by each term inside the parentheses.
  • Simplifying radicals involves applying the rules of radicals to simplify the resulting expression.
  • Factoring out the perfect square from the radicand can help simplify the radical expression further.
  • Combining the simplified expressions can help get the final result.

Practice Problems

  1. Distribute and simplify the radical expression: 18⋅(−2+3)\sqrt{18} \cdot (-2+\sqrt{3})
  2. Distribute and simplify the radical expression: 24⋅(2−2)\sqrt{24} \cdot (2-\sqrt{2})
  3. Distribute and simplify the radical expression: 36⋅(3+2)\sqrt{36} \cdot (3+\sqrt{2})

Answer Key

  1. −62+36\boxed{-6\sqrt{2} + 3\sqrt{6}}
  2. −43+46\boxed{-4\sqrt{3} + 4\sqrt{6}}
  3. 62+63\boxed{6\sqrt{2} + 6\sqrt{3}}
    Distribute and Simplify Radicals: Q&A =====================================

Q: What is the distributive property of radicals?

A: The distributive property of radicals states that for any real numbers aa, bb, and cc, a⋅(b+c)=a⋅b+a⋅c\sqrt{a} \cdot (b+c) = \sqrt{a} \cdot b + \sqrt{a} \cdot c. This means that we can distribute the radical expression to each term inside the parentheses.

Q: How do I distribute radicals to each term inside the parentheses?

A: To distribute radicals to each term inside the parentheses, we need to multiply the radical expression by each term. For example, if we have the expression 12⋅(−1+5)\sqrt{12} \cdot (-1+\sqrt{5}), we would multiply the radical expression by each term inside the parentheses: 12⋅(−1)+12⋅5\sqrt{12} \cdot (-1) + \sqrt{12} \cdot \sqrt{5}.

Q: What are the rules of radicals?

A: The rules of radicals state that for any real numbers aa and bb, ab=a⋅b\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}. This means that we can simplify the radical expression by factoring out the perfect square from the radicand.

Q: How do I simplify the radical expression by factoring out the perfect square?

A: To simplify the radical expression by factoring out the perfect square, we need to identify the perfect square that can be factored out from the radicand. For example, if we have the expression 60\sqrt{60}, we can factor out 44 from 6060, which is a perfect square: 60=4⋅15=4⋅15=215\sqrt{60} = \sqrt{4 \cdot 15} = \sqrt{4} \cdot \sqrt{15} = 2\sqrt{15}.

Q: Can I simplify the radical expression further?

A: Yes, we can simplify the radical expression further by factoring out the perfect square from the radicand. For example, if we have the expression −12-\sqrt{12}, we can factor out 44 from 1212, which is a perfect square: −12=−4⋅3=−4⋅3=−23-\sqrt{12} = -\sqrt{4 \cdot 3} = -\sqrt{4} \cdot \sqrt{3} = -2\sqrt{3}.

Q: How do I combine the simplified expressions to get the final result?

A: To combine the simplified expressions, we need to add or subtract the simplified expressions. For example, if we have the expressions −23-2\sqrt{3} and 2152\sqrt{15}, we can combine them by adding them: −23+215-2\sqrt{3} + 2\sqrt{15}.

Q: What are some common mistakes to avoid when distributing and simplifying radicals?

A: Some common mistakes to avoid when distributing and simplifying radicals include:

  • Not distributing the radical expression to each term inside the parentheses
  • Not simplifying the radical expression by factoring out the perfect square
  • Not combining the simplified expressions to get the final result
  • Not checking for errors in the simplification process

Q: How can I practice distributing and simplifying radicals?

A: You can practice distributing and simplifying radicals by working through practice problems, such as:

  • Distributing and simplifying the radical expression: 18⋅(−2+3)\sqrt{18} \cdot (-2+\sqrt{3})
  • Distributing and simplifying the radical expression: 24⋅(2−2)\sqrt{24} \cdot (2-\sqrt{2})
  • Distributing and simplifying the radical expression: 36⋅(3+2)\sqrt{36} \cdot (3+\sqrt{2})

Q: What are some real-world applications of distributing and simplifying radicals?

A: Distributing and simplifying radicals have many real-world applications, such as:

  • Calculating the area and perimeter of shapes with radical dimensions
  • Solving problems involving the motion of objects with radical velocities
  • Working with electrical circuits that involve radical resistances

Q: Can I use a calculator to simplify radicals?

A: Yes, you can use a calculator to simplify radicals. However, it's always a good idea to check your work by hand to ensure that you get the correct answer.

Q: How can I check my work when simplifying radicals?

A: You can check your work by:

  • Verifying that you distributed the radical expression to each term inside the parentheses
  • Verifying that you simplified the radical expression by factoring out the perfect square
  • Verifying that you combined the simplified expressions to get the final result
  • Checking for errors in the simplification process