What Are The Square Roots Of 64? Check All That Apply.A. { |8|$}$ B. 4 C. -8 D. 8 E. -4 F. 8.5
What are the Square Roots of 64? Check All That Apply
In mathematics, the square root of a number is a value that, when multiplied by itself, gives the original number. In this article, we will explore the square roots of 64 and determine which of the given options are correct.
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.
To find the square roots of 64, we need to determine which values, when multiplied by themselves, give 64. We can start by listing the perfect squares that are less than or equal to 64.
- 1 × 1 = 1
- 2 × 2 = 4
- 3 × 3 = 9
- 4 × 4 = 16
- 5 × 5 = 25
- 6 × 6 = 36
- 7 × 7 = 49
- 8 × 8 = 64
As we can see, 8 is a square root of 64, because 8 multiplied by 8 equals 64. However, we also need to consider the negative square root of 64.
- -8 × -8 = 64
Therefore, both 8 and -8 are square roots of 64.
Now that we have determined the square roots of 64, let's evaluate the given options.
A. {|8|$}$: This option is incorrect, because the absolute value of 8 is 8, but we are looking for the square roots of 64, not the absolute value of 8.
B. 4: This option is incorrect, because 4 is not a square root of 64.
C. -8: This option is correct, because -8 is a square root of 64.
D. 8: This option is correct, because 8 is a square root of 64.
E. -4: This option is incorrect, because -4 is not a square root of 64.
F. 8.5: This option is incorrect, because 8.5 is not a square root of 64.
In conclusion, the square roots of 64 are 8 and -8. Therefore, the correct options are C and D.
When working with square roots, it's essential to remember that both positive and negative values are considered square roots of a number. Additionally, the square root of a number can be a rational or irrational number, depending on the number.
When evaluating square roots, it's common to make mistakes such as:
- Assuming that only positive values are square roots of a number.
- Failing to consider the negative square root of a number.
- Not checking if a value is a perfect square before determining its square root.
Square roots have numerous real-world applications, including:
- Calculating distances and heights in geometry and trigonometry.
- Determining the area and perimeter of shapes in geometry.
- Calculating the speed and distance of objects in physics.
- Analyzing data and statistics in finance and economics.
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: How do I find the square root of a number?
A: To find the square root of a number, you can use a calculator or a mathematical formula. The formula for finding the square root of a number is:
√x = y, where x is the number and y is the square root of x.
Q: What are the square roots of 64?
A: The square roots of 64 are 8 and -8, because 8 multiplied by 8 equals 64, and -8 multiplied by -8 also equals 64.
Q: Why are there two square roots of a number?
A: There are two square roots of a number because the square root of a number can be positive or negative. For example, the square root of 16 is both 4 and -4, because 4 multiplied by 4 equals 16, and -4 multiplied by -4 also equals 16.
Q: How do I determine if a number is a perfect square?
A: To determine if a number is a perfect square, you can check if it can be expressed as the product of two equal integers. For example, 16 is a perfect square because it can be expressed as 4 × 4, and 64 is also a perfect square because it can be expressed as 8 × 8.
Q: What are some real-world applications of square roots?
A: Square roots have numerous real-world applications, including:
- Calculating distances and heights in geometry and trigonometry.
- Determining the area and perimeter of shapes in geometry.
- Calculating the speed and distance of objects in physics.
- Analyzing data and statistics in finance and economics.
Q: How do I use a calculator to find the square root of a number?
A: To use a calculator to find the square root of a number, follow these steps:
- Enter the number you want to find the square root of.
- Press the square root button (usually denoted by √).
- The calculator will display the square root of the number.
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 is the result of multiplying a number by itself, while a square root is the value that, when multiplied by itself, gives the original number. For example, the square of 4 is 16, while the square root of 16 is 4.
Q: Can I use square roots to solve equations?
A: Yes, you can use square roots to solve equations. For example, if you have the equation x^2 = 16, you can solve for x by taking the square root of both sides of the equation, which gives you x = ±4.
Q: What are some common mistakes to avoid when working with square roots?
A: Some common mistakes to avoid when working with square roots include:
- Assuming that only positive values are square roots of a number.
- Failing to consider the negative square root of a number.
- Not checking if a value is a perfect square before determining its square root.
Q: How do I simplify square roots?
A: To simplify square roots, you can use the following steps:
- Check if the number under the square root sign is a perfect square.
- If it is, you can simplify the square root by taking the square root of the perfect square.
- If it is not, you can leave the square root as is.
Q: Can I use square roots to solve problems in other areas of mathematics?
A: Yes, you can use square roots to solve problems in other areas of mathematics, including algebra, geometry, and trigonometry. Square roots are a fundamental concept in mathematics and have numerous applications in various areas of mathematics.