Select The Correct Choice Below And, If Necessary, Fill In The Answer Box Within Your Choice.A. $\sqrt[3]{1000} =$ $\square$ B. The Root Is Not A Real Number.

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Introduction

In mathematics, the cube root of a number is a value that, when multiplied by itself twice, gives the original number. It is denoted by the symbol x3\sqrt[3]{x}, where xx is the number inside the radical sign. In this article, we will explore the concept of cube roots and real numbers, and determine the correct choice for the equation 10003=\sqrt[3]{1000} =.

What are Cube Roots?

A cube root is a number that, when multiplied by itself twice, gives the original number. For example, the cube root of 27 is 3, because 3×3×3=273 \times 3 \times 3 = 27. Cube roots are denoted by the symbol x3\sqrt[3]{x}, where xx is the number inside the radical sign.

Real Numbers

Real numbers are numbers that can be expressed on the number line. They include all rational and irrational numbers. Rational numbers are numbers that can be expressed as the ratio of two integers, such as 3/4 or 22/7. Irrational numbers are numbers that cannot be expressed as a ratio of two integers, such as the square root of 2 or the cube root of 3.

Cube Roots of Perfect Cubes

Perfect cubes are numbers that can be expressed as the cube of an integer. For example, 27 is a perfect cube because it can be expressed as 3×3×33 \times 3 \times 3. The cube root of a perfect cube is always a real number. For example, the cube root of 27 is 3, because 3×3×3=273 \times 3 \times 3 = 27.

Cube Roots of Non-Perfect Cubes

Non-perfect cubes are numbers that cannot be expressed as the cube of an integer. For example, 1000 is a non-perfect cube because it cannot be expressed as the cube of an integer. The cube root of a non-perfect cube may or may not be a real number.

Determining the Correct Choice

Now that we have a basic understanding of cube roots and real numbers, let's determine the correct choice for the equation 10003=\sqrt[3]{1000} =.

  • Choice A: 10003=\sqrt[3]{1000} = â–¡\square
  • Choice B: The root is not a real number.

To determine the correct choice, we need to find the cube root of 1000. We can do this by using a calculator or by finding the cube root of 1000 by hand.

Finding the Cube Root of 1000

To find the cube root of 1000, we can use a calculator or by finding the cube root of 1000 by hand. Using a calculator, we get:

10003=10\sqrt[3]{1000} = 10

Therefore, the correct choice is:

  • Choice A: 10003=\sqrt[3]{1000} = 10

Conclusion

In conclusion, the cube root of 1000 is 10, which is a real number. Therefore, the correct choice is Choice A: 10003=\sqrt[3]{1000} = 10.

Additional Examples

Here are some additional examples of cube roots and real numbers:

  • Example 1: Find the cube root of 64.
  • Example 2: Find the cube root of 125.
  • Example 3: Find the cube root of 216.

Solutions

  • Example 1: The cube root of 64 is 4, because 4×4×4=644 \times 4 \times 4 = 64.
  • Example 2: The cube root of 125 is 5, because 5×5×5=1255 \times 5 \times 5 = 125.
  • Example 3: The cube root of 216 is 6, because 6×6×6=2166 \times 6 \times 6 = 216.

Conclusion

In conclusion, cube roots are an important concept in mathematics that can be used to solve a variety of problems. By understanding the concept of cube roots and real numbers, we can determine the correct choice for the equation 10003=\sqrt[3]{1000} =.

References

  • "Cube Root" by Math Open Reference
  • "Real Numbers" by Khan Academy
  • "Cube Roots of Perfect Cubes" by Purplemath
  • "Cube Roots of Non-Perfect Cubes" by Mathway

Final Answer

Introduction

In our previous article, we explored the concept of cube roots and real numbers, and determined the correct choice for the equation 10003=\sqrt[3]{1000} =.

Q&A

Q: What is a cube root?

A: A cube root is a number that, when multiplied by itself twice, gives the original number. It is denoted by the symbol x3\sqrt[3]{x}, where xx is the number inside the radical sign.

Q: What are real numbers?

A: Real numbers are numbers that can be expressed on the number line. They include all rational and irrational numbers. Rational numbers are numbers that can be expressed as the ratio of two integers, such as 3/4 or 22/7. Irrational numbers are numbers that cannot be expressed as a ratio of two integers, such as the square root of 2 or the cube root of 3.

Q: What is the difference between a perfect cube and a non-perfect cube?

A: A perfect cube is a number that can be expressed as the cube of an integer. For example, 27 is a perfect cube because it can be expressed as 3×3×33 \times 3 \times 3. A non-perfect cube is a number that cannot be expressed as the cube of an integer. For example, 1000 is a non-perfect cube because it cannot be expressed as the cube of an integer.

Q: How do I determine if a cube root is a real number or not?

A: To determine if a cube root is a real number or not, you need to check if the number inside the radical sign is a perfect cube or not. If it is a perfect cube, then the cube root is a real number. If it is not a perfect cube, then the cube root may or may not be a real number.

Q: Can you give me some examples of cube roots and real numbers?

A: Here are some examples:

  • Example 1: Find the cube root of 64.
  • Example 2: Find the cube root of 125.
  • Example 3: Find the cube root of 216.

Solutions

  • Example 1: The cube root of 64 is 4, because 4×4×4=644 \times 4 \times 4 = 64.
  • Example 2: The cube root of 125 is 5, because 5×5×5=1255 \times 5 \times 5 = 125.
  • Example 3: The cube root of 216 is 6, because 6×6×6=2166 \times 6 \times 6 = 216.

Q: What are some common mistakes to avoid when working with cube roots and real numbers?

A: Here are some common mistakes to avoid:

  • Mistake 1: Assuming that all cube roots are real numbers.
  • Mistake 2: Not checking if the number inside the radical sign is a perfect cube or not.
  • Mistake 3: Not using a calculator or a computer program to check if the cube root is a real number or not.

Conclusion

In conclusion, cube roots and real numbers are important concepts in mathematics that can be used to solve a variety of problems. By understanding the concept of cube roots and real numbers, we can determine the correct choice for the equation 10003=\sqrt[3]{1000} =.

References

  • "Cube Root" by Math Open Reference
  • "Real Numbers" by Khan Academy
  • "Cube Roots of Perfect Cubes" by Purplemath
  • "Cube Roots of Non-Perfect Cubes" by Mathway

Final Answer

The final answer is: 10\boxed{10}