Which Of The Following Is Equivalent To A Real Number?A. ( − 144 ) 1 / 3 (-144)^{1 / 3} ( − 144 ) 1/3 B. ( − 196 ) 1 / 4 (-196)^{1 / 4} ( − 196 ) 1/4 C. ( − 1024 ) 1 / 2 (-1024)^{1 / 2} ( − 1024 ) 1/2 D. ( − 1503 ) 1 / 6 (-1503)^{1 / 6} ( − 1503 ) 1/6

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Which of the Following is Equivalent to a Real Number?

Understanding the Concept of Real Numbers

Real numbers are a set of numbers that include all rational and irrational numbers. They are used to represent quantities that can be measured or expressed on a number line. Real numbers can be positive, negative, or zero, and they can be expressed in decimal or fraction form. In mathematics, real numbers play a crucial role in various mathematical operations, including addition, subtraction, multiplication, and division.

Analyzing the Options

To determine which of the given options is equivalent to a real number, we need to analyze each option separately. We will use the properties of exponents and roots to simplify each expression and determine if it results in a real number.

Option A: (−144)1/3(-144)^{1 / 3}

To simplify this expression, we need to find the cube root of -144. The cube root of a negative number is a real number if the index of the root is odd. In this case, the index is 3, which is an odd number. Therefore, the cube root of -144 is a real number.

(-144)^{1 / 3} = -\sqrt[3]{144} = -\sqrt[3]{12^2} = -12

Option B: (−196)1/4(-196)^{1 / 4}

To simplify this expression, we need to find the fourth root of -196. The fourth root of a negative number is a real number if the index of the root is even. In this case, the index is 4, which is an even number. Therefore, the fourth root of -196 is a real number.

(-196)^{1 / 4} = -\sqrt[4]{196} = -\sqrt[4]{14^2} = -14

Option C: (−1024)1/2(-1024)^{1 / 2}

To simplify this expression, we need to find the square root of -1024. The square root of a negative number is not a real number. In mathematics, the square root of a negative number is an imaginary number, which is a complex number that cannot be expressed on a number line.

(-1024)^{1 / 2} = \sqrt{-1024} = \sqrt{-32^2} = 32i

Option D: (−1503)1/6(-1503)^{1 / 6}

To simplify this expression, we need to find the sixth root of -1503. The sixth root of a negative number is not a real number. In mathematics, the sixth root of a negative number is an imaginary number, which is a complex number that cannot be expressed on a number line.

(-1503)^{1 / 6} = \sqrt[6]{-1503} = \sqrt[6]{-3^2 \cdot 167} = -\sqrt[6]{3^2 \cdot 167} = -\sqrt[3]{\sqrt[3]{3^2 \cdot 167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[3]{3 \cdot \sqrt[3]{167}} = -\sqrt[<br/>
**Which of the Following is Equivalent to a Real Number? - Q&A**

Understanding the Concept of Real Numbers


Real numbers are a set of numbers that include all rational and irrational numbers. They are used to represent quantities that can be measured or expressed on a number line. Real numbers can be positive, negative, or zero, and they can be expressed in decimal or fraction form. In mathematics, real numbers play a crucial role in various mathematical operations, including addition, subtraction, multiplication, and division.


Q: What is the difference between a real number and an imaginary number?


A: A real number is a number that can be expressed on a number line, while an imaginary number is a complex number that cannot be expressed on a number line. Imaginary numbers are used to represent quantities that are not real, such as the square root of a negative number.


Q: Can you give an example of a real number?


A: Yes, some examples of real numbers include:


    

  • Positive numbers: 1, 2, 3, ...
    • 

  • Negative numbers: -1, -2, -3, ...
    • 

  • Zero: 0
    • 

  • Fractions: 1/2, 3/4, 2/3, ...
    • 

  • Decimals: 0.5, 0.25, 0.75, ...
    • 



Q: Can you give an example of an imaginary number?


A: Yes, some examples of imaginary numbers include:


    

  • The square root of a negative number: √(-1), √(-4), √(-9), ...
    • 

  • Complex numbers: 3 + 4i, 2 - 5i, 1 + 2i, ...
    • 



Q: How do you determine if a number is real or imaginary?


A: To determine if a number is real or imaginary, you need to check if it can be expressed on a number line. If it can be expressed on a number line, it is a real number. If it cannot be expressed on a number line, it is an imaginary number.


Q: Can you give an example of a real number that is equivalent to one of the options in the original question?


A: Yes, one example of a real number that is equivalent to one of the options in the original question is:


    

  • Option A: (−144)1/3=−12(-144)^{1 / 3} = -12
    • 



This is a real number because the cube root of a negative number is a real number if the index of the root is odd.


Q: Can you give an example of an imaginary number that is equivalent to one of the options in the original question?


A: Yes, one example of an imaginary number that is equivalent to one of the options in the original question is:


    

  • Option C: (−1024)1/2=32i(-1024)^{1 / 2} = 32i
    • 



This is an imaginary number because the square root of a negative number is an imaginary number.


Q: What is the significance of real numbers in mathematics?


A: Real numbers play a crucial role in various mathematical operations, including addition, subtraction, multiplication, and division. They are used to represent quantities that can be measured or expressed on a number line. Real numbers are also used in various mathematical concepts, such as algebra, geometry, and calculus.


Q: Can you give an example of a real number that is used in a mathematical operation?


A: Yes, one example of a real number that is used in a mathematical operation is:


    

  • 2 + 3 = 5
    • 



In this example, the real numbers 2 and 3 are added together to get the result 5.


Q: Can you give an example of an imaginary number that is used in a mathematical operation?


A: Yes, one example of an imaginary number that is used in a mathematical operation is:


    

  • 3 + 4i
    • 



In this example, the imaginary number 4i is added to the real number 3 to get the result 3 + 4i.


Q: What is the difference between a real number and a complex number?


A: A real number is a number that can be expressed on a number line, while a complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit. Complex numbers are used to represent quantities that are not real, such as the square root of a negative number.


Q: Can you give an example of a complex number?


A: Yes, one example of a complex number is:


    

  • 3 + 4i
    • 



In this example, the complex number 3 + 4i is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit.


Q: Can you give an example of a real number that is equivalent to a complex number?


A: Yes, one example of a real number that is equivalent to a complex number is:


    

  • 3 + 0i = 3
    • 



In this example, the real number 3 is equivalent to the complex number 3 + 0i.