Select The Correct Answer.What Is This Expression In Simplified Form? − 864 3 \sqrt[3]{-864} 3 − 864 ​ A. − 4 6 3 -4 \sqrt[3]{6} − 4 3 6 ​ B. − 6 4 3 -6 \sqrt[3]{4} − 6 3 4 ​ C. − 12 6 3 -12 \sqrt[3]{6} − 12 3 6 ​ D. − 6 12 3 -6 \sqrt[3]{12} − 6 3 12 ​

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Introduction

Radicals 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 radicals, with a focus on the expression −8643\sqrt[3]{-864}. We will break down the steps involved in simplifying this expression and provide a clear explanation of the reasoning behind each step.

Understanding Radicals

A radical is a mathematical expression that represents a number that can be expressed as the product of a perfect square and another number. The radical symbol, xn\sqrt[n]{x}, is used to denote the nth root of a number x. In the case of the expression −8643\sqrt[3]{-864}, we are dealing with a cube root, which means that we are looking for a number that, when cubed, gives us -864.

Breaking Down the Expression

To simplify the expression −8643\sqrt[3]{-864}, we need to break it down into its prime factors. This involves finding the prime numbers that multiply together to give us -864. We can start by factoring out the negative sign, which gives us:

−8643=−24⋅333\sqrt[3]{-864} = \sqrt[3]{-2^4 \cdot 3^3}

Simplifying the Radical

Now that we have broken down the expression into its prime factors, we can simplify the radical by taking out any perfect cubes. In this case, we can take out a perfect cube of 3, which gives us:

−24⋅333=−21⋅31⋅21⋅303\sqrt[3]{-2^4 \cdot 3^3} = -2^1 \cdot 3^1 \cdot \sqrt[3]{2^1 \cdot 3^0}

Evaluating the Expression

Now that we have simplified the radical, we can evaluate the expression by multiplying the numbers together. This gives us:

−21⋅31⋅21⋅303=−2⋅3⋅23=−623-2^1 \cdot 3^1 \cdot \sqrt[3]{2^1 \cdot 3^0} = -2 \cdot 3 \cdot \sqrt[3]{2} = -6 \sqrt[3]{2}

Conclusion

In conclusion, the expression −8643\sqrt[3]{-864} can be simplified to −623-6 \sqrt[3]{2}. This involves breaking down the expression into its prime factors, taking out any perfect cubes, and then evaluating the resulting expression.

Comparison with Answer Choices

Now that we have simplified the expression, we can compare it with the answer choices provided. The correct answer is:

  • A. −463-4 \sqrt[3]{6}: This is not the correct answer, as the simplified expression is −623-6 \sqrt[3]{2}, not −463-4 \sqrt[3]{6}.
  • B. −643-6 \sqrt[3]{4}: This is not the correct answer, as the simplified expression is −623-6 \sqrt[3]{2}, not −643-6 \sqrt[3]{4}.
  • C. −1263-12 \sqrt[3]{6}: This is not the correct answer, as the simplified expression is −623-6 \sqrt[3]{2}, not −1263-12 \sqrt[3]{6}.
  • D. −6123-6 \sqrt[3]{12}: This is not the correct answer, as the simplified expression is −623-6 \sqrt[3]{2}, not −6123-6 \sqrt[3]{12}.

Final Answer

The correct answer is B. −643-6 \sqrt[3]{4} is incorrect, the correct answer is −623-6 \sqrt[3]{2}

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Introduction

In our previous article, we explored the process of simplifying radicals, with a focus on the expression −8643\sqrt[3]{-864}. We broke down the steps involved in simplifying this expression and provided a clear explanation of the reasoning behind each step. In this article, we will continue to explore the topic of simplifying radicals, this time in the form of a Q&A guide.

Q: What is a radical?

A: A radical is a mathematical expression that represents a number that can be expressed as the product of a perfect square and another number. The radical symbol, xn\sqrt[n]{x}, is used to denote the nth root of a number x.

Q: How do I simplify a radical?

A: To simplify a radical, you need to break it down into its prime factors. This involves finding the prime numbers that multiply together to give you the number inside the radical. You can then take out any perfect squares or cubes, and simplify the resulting expression.

Q: What is the difference between a square root and a cube root?

A: A square root is the nth root of a number, where n is 2. A cube root is the nth root of a number, where n is 3. In other words, a square root is the number that, when squared, gives you the original number, while a cube root is the number that, when cubed, gives you the original number.

Q: How do I simplify a cube root?

A: To simplify a cube root, you need to break it down into its prime factors. This involves finding the prime numbers that multiply together to give you the number inside the cube root. You can then take out any perfect cubes, and simplify the resulting expression.

Q: What is the formula for simplifying a cube root?

A: The formula for simplifying a cube root is:

a3⋅b3=a⋅b3\sqrt[3]{a^3 \cdot b} = a \cdot \sqrt[3]{b}

where a is a perfect cube and b is any number.

Q: How do I evaluate a cube root?

A: To evaluate a cube root, you need to multiply the numbers together. This involves multiplying the number inside the cube root by the cube root itself.

Q: What is the difference between a cube root and a square root?

A: A cube root is the nth root of a number, where n is 3. A square root is the nth root of a number, where n is 2. In other words, a cube root is the number that, when cubed, gives you the original number, while a square root is the number that, when squared, gives you the original number.

Q: How do I simplify a radical with a negative number?

A: To simplify a radical with a negative number, you need to break it down into its prime factors. This involves finding the prime numbers that multiply together to give you the number inside the radical. You can then take out any perfect squares or cubes, and simplify the resulting expression.

Q: What is the formula for simplifying a radical with a negative number?

A: The formula for simplifying a radical with a negative number is:

−a2⋅bn=−a⋅bn\sqrt[n]{-a^2 \cdot b} = -a \cdot \sqrt[n]{b}

where a is a perfect square and b is any number.

Q: How do I evaluate a radical with a negative number?

A: To evaluate a radical with a negative number, you need to multiply the numbers together. This involves multiplying the number inside the radical by the radical itself.

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

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

  • Not breaking down the number inside the radical into its prime factors
  • Not taking out any perfect squares or cubes
  • Not simplifying the resulting expression
  • Not evaluating the expression correctly

Q: How can I practice simplifying radicals?

A: You can practice simplifying radicals by working through examples and exercises. You can also use online resources, such as calculators and worksheets, to help you practice.

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

A: Simplifying radicals has many real-world applications, including:

  • Calculating distances and heights in geometry and trigonometry
  • Solving equations and inequalities in algebra
  • Working with complex numbers and polynomials
  • Calculating probabilities and statistics in data analysis

Q: How can I use simplifying radicals in my everyday life?

A: Simplifying radicals can be used in many everyday situations, including:

  • Calculating the area and perimeter of a room or building
  • Determining the height of a building or object
  • Calculating the distance between two points
  • Solving problems in finance and economics

Conclusion

In conclusion, simplifying radicals is a crucial skill to master in mathematics. By understanding the process of simplifying radicals, you can apply it to a wide range of real-world situations. Remember to break down the number inside the radical into its prime factors, take out any perfect squares or cubes, and simplify the resulting expression. With practice and patience, you can become proficient in simplifying radicals and apply it to your everyday life.