Which Of The Following Are All Square Roots Of 64?A. $(-64)^{\frac{1}{2}}, -8, 8$B. 4, 8, 16C. $64^{\frac{1}{2}}, -8, 8$D. 8, 16, 32

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Introduction

In mathematics, square roots are a fundamental concept that plays a crucial role in various mathematical operations and applications. A square root of a number is a value that, when multiplied by itself, gives the original number. In this article, we will explore the concept of square roots and identify the correct square roots of 64 from the given options.

What are Square Roots?

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because 4 multiplied by 4 equals 16. The square root of a number can be positive or negative, and both values are considered square roots of the number.

Properties of Square Roots

There are several properties of square roots that are essential to understand:

  • The square root of a number is always non-negative.
  • The square root of a number can be positive or negative.
  • The square root of a number is denoted by the symbol √.
  • The square root of a number can be expressed as a decimal or a fraction.

Finding Square Roots of 64

To find the square roots of 64, we need to identify the values that, when multiplied by themselves, give 64. Let's analyze the given options:

Option A: (−64)12,−8,8(-64)^{\frac{1}{2}}, -8, 8

To determine if these values are square roots of 64, we need to check if they satisfy the condition of being a square root. The value (−64)12(-64)^{\frac{1}{2}} is equal to -8, which is a square root of 64. However, we need to check if the other values in this option are also square roots of 64.

Option B: 4, 8, 16

To determine if these values are square roots of 64, we need to check if they satisfy the condition of being a square root. The value 4 multiplied by 4 equals 16, not 64. Therefore, 4 is not a square root of 64. The value 8 multiplied by 8 equals 64, so 8 is a square root of 64. However, we need to check if the other values in this option are also square roots of 64.

Option C: 6412,−8,864^{\frac{1}{2}}, -8, 8

To determine if these values are square roots of 64, we need to check if they satisfy the condition of being a square root. The value 641264^{\frac{1}{2}} is equal to 8, which is a square root of 64. We have already established that -8 is also a square root of 64. Therefore, this option contains two square roots of 64.

Option D: 8, 16, 32

To determine if these values are square roots of 64, we need to check if they satisfy the condition of being a square root. The value 8 multiplied by 8 equals 64, so 8 is a square root of 64. However, we need to check if the other values in this option are also square roots of 64.

Conclusion

In conclusion, the correct square roots of 64 are (−64)12,−8,8(-64)^{\frac{1}{2}}, -8, 8 and 6412,−8,864^{\frac{1}{2}}, -8, 8. These options contain two square roots of 64, which are -8 and 8. The other options do not contain all the square roots of 64.

Final Answer

The final answer is:

  • Option A: (−64)12,−8,8(-64)^{\frac{1}{2}}, -8, 8
  • Option C: 6412,−8,864^{\frac{1}{2}}, -8, 8
    Understanding Square Roots and Their Applications =====================================================

Q&A: Square Roots and Their Applications

Q: What is a square root?

A: A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because 4 multiplied by 4 equals 16.

Q: What are the properties of square roots?

A: There are several properties of square roots that are essential to understand:

  • The square root of a number is always non-negative.
  • The square root of a number can be positive or negative.
  • The square root of a number is denoted by the symbol √.
  • The square root of a number can be expressed as a decimal or a fraction.

Q: How do I find the square roots of a number?

A: To find the square roots of a number, you need to identify the values that, when multiplied by themselves, give the original number. You can use a calculator or a mathematical formula to find the square roots of a number.

Q: What are the square roots of 64?

A: The square roots of 64 are -8 and 8. These values, when multiplied by themselves, give 64.

Q: Can a square root be a decimal or a fraction?

A: Yes, a square root can be a decimal or a fraction. For example, the square root of 2 is approximately 1.414, which is a decimal. The square root of 9 is 3, which is a whole number. The square root of 4 is 2, which is a whole number.

Q: How do I simplify a square root?

A: To simplify a square root, you need to find the largest perfect square that divides the number inside the square root. For example, the square root of 18 can be simplified as √(9 * 2) = 3√2.

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

A: A square root and a square are related but distinct concepts. A square root is a value that, when multiplied by itself, gives the original number. A square is the result of multiplying a number by itself. For example, the square root of 16 is 4, and the square of 4 is 16.

Q: Can a square root be a negative number?

A: Yes, a square root can be a negative number. For example, the square root of -16 is -4, and the square of -4 is -16.

Q: How do I use square roots in real-life applications?

A: Square roots are used in various real-life applications, such as:

  • Calculating distances and heights
  • Determining the area and perimeter of shapes
  • Finding the volume of 3D objects
  • Solving equations and inequalities

Conclusion

In conclusion, square roots are a fundamental concept in mathematics that plays a crucial role in various mathematical operations and applications. Understanding the properties and applications of square roots can help you solve problems and make informed decisions in your personal and professional life.

Final Answer

The final answer is:

  • The square roots of 64 are -8 and 8.
  • A square root can be a decimal or a fraction.
  • A square root can be simplified by finding the largest perfect square that divides the number inside the square root.
  • A square root and a square are related but distinct concepts.
  • A square root can be a negative number.
  • Square roots are used in various real-life applications, such as calculating distances and heights, determining the area and perimeter of shapes, finding the volume of 3D objects, and solving equations and inequalities.